Munich

LINEAR SYSTEMS OF QUADRICS. MÜNICH.
Oct.-Dec. 2016
Igor Dolgachev
May 24, 2020

2
Contents
3
4 CONTENTS
Lecture 1
Quadrics
1.1 Quadratic forms

Here we recall some standard facts and notations from the theory of quadratic
forms that can be found in a good text-book in linear algebra. We let k to
be a field of characteristic 6= 2 and refer to the Appendix for the case when
the characteristic is equal to 2.
Let E be a linear space of dimension n + 1 over k. A quadratic form q
on E is an element of the second symmetric square S 2 E ∨ of the dual linear
space E ∨ . If we choose a basis e1 , . . . , en+1 in E, then q is given by an
expression X
q= aij ti tj , (1.1.1)
1≤i≤j≤n+1
where t1 , . . . , tn+1 is the dual basis in E ∨ (it serves as coordinates in E)

and aij ∈ k. A quadratic form can be considered as a function q : E → k
characterized by the properties
(a) q(λv) = λ2 q(v), for any λ ∈ k and v ∈ E;
(b) the function bq : E × E → k : (v, w) 7→ q(v + w) − q(v) − q(w), is

a symmetric bilinear form on E, i.e. an element of the linear space
(S 2 E)∨ ⊂ E ∨ ⊗ E ∨ .
The symmetric bilinear form bq is called the polar bilinear form associated
with q. Its expression in the basis (ti ⊗ tj )1≤i≤j≤n+1 of E ∨ ⊗ E ∨ is given by
n+1
X X
bq = 2 aii ti ⊗ ti + aij (ti ⊗ tj + tj ⊗ ti ).
i=1 1≤i<j≤n+1
5
6 LECTURE 1. QUADRICS
The symmetric matrix A with diagonal elements 2aii and off-diagonal ele-
ments aij = aji is called the matrix of bq in the basis (e1 , . . . , en+1 ). We have
bq (ei , ej ) = bq (ej , ei ) = aij if i < j and bq (ei , ei ) = 2aii otherwise. Since the
characteristic is different from two, q(x) = 12 bq (x, x), so a quadratic form is
completely determined by the polar bilinear form.
As any bilinear form, the polar bilinear form bq can be identified with
a linear map E → E ∨ that assigns to a vector v the linear function w 7→
bq (v, w). If we identify (E ∨ )∨ with E, then the symmetry condition is equiv-
alent to that the linear map coincides with its transpose map. The matrix
of this map in a basis of E and its dual basis of E ∨ is the matrix A of bq
defined in above. Its determinant is defined uniquely up to squares in k and
is called the discriminant of q.
The kernel of the linear function bq : E → E ∨ is called the radical of bq
and will be denoted by rad(bq ). Its dimension is called the corank or the
defect of q and of bq and the dimension of its image is called the rank of q
or of bq .
A quadratic form q is called degenerate if its defect δ(q) is not equal to
zero. A non-zero vector v in the radical rad(bq ) of bq is called a singular
vector.
For any subset L of E, we denote by L⊥ the orthogonal complement of
L with respect to bq , i.e
L⊥ := {v ∈ E : bq (v, w) = 0, for any w ∈ L}. (1.1.2)
This is obviously a linear subspace of E. For example, rad(bq ) = E ⊥ and

for any L, we have rad(bq ) ⊂ L⊥ . Note that, if q is non-degenerate, and L
is a subspace, then
dim L⊥ = dim E − dim L.
Let q be a totally degenerate P quadratic form, i.e. rad(bq ) = 0. If p 6= 2,

then q = 0. Otherwise q = 2
ai xi in some basis in E. In some radical
extension K/k and rad(bq ) be the radical of bq . Choose a basis e1 , . . . , ec
of rad(bq ) and let L be the complementary subspace of rad(bq ). Then there
exists a finite extension K/k such that q vanishes on some nonzero vector
v1 ∈ LK . (Prove that one may choose a separable extension even
when char(k) = 2). Let w1 ∈ LK be an element from v1⊥ . Since the radical
of bq restricted to L is trivial, we can choose w1 with bq (v1 , w1 ) = 1. Let H1
be the plane spanned by v1 and w1 , and E 0 = H ⊥ be its orthogonal comple-
ment in EK . Arguing, by induction on n, we find a basis (v1 , w1 , . . . , vm , wm )
if n + 1 = 2m and (v1 , w1 , . . . , vm , wm , e) if n = 2m such that in coordinates
1.1. QUADRATIC FORMS 7
defined by this basis the expression of q is given by

m
X
q = t2i t2i−1 if n = 2m − 1, (1.1.3)
i=1
m
X
q = t2i t2i−1 + t22m+1 if n = 2m. (1.1.4)
i=1
A vector v ∈ E is called an isotropic vector of a quadratic form q if

q(v) = 0. A linear subspace F in E such that the restriction of bq to F is
identical zero is called a totally isotropic subspace. Since p = char(k) 6= 0,
then F is totally isotropic if and only if the restriction of q to F is zero.
An example of a totally isotropic subspace of q is its radical rad(bq ). If L
is isotropic, we have L ⊂ L⊥ . If q is non-degenerate, applying (??), we
find that dim L ≤ [ n+12 ]. It follows from the formulas (??) that there exists
an isotropic subspace of dimension [ n+1 2 ]. It is defined by the equations
t2 = . . . = t2m = 0 and t2m+1 if n is even. A totally isotropic subspace of
this dimension is called a maximal isotropic subspace.
By a theorem of Witt, any totally isotropic subspace is contained in a
maximal totally isotropic subspace of dimension m = [ 12 (n + 1)] (if q is non-
degenerate). Two totally isotropic subspaces of the same dimension can be
transformed to each other by an element of the orthogonal group O(E; q) of
linear transformations of E that preserve the quadratic form q.
Recall that the orthogonal group O(E, q) ∼ = Ok (n + 1) is an algebraic
group of dimension n(n + 1)/2. It consists of two connected components,
the connected component of the identity is the group SOk (n + 1). The latter
is not simply-connected if n is odd, its universal cover is the spin group
Spink (n + 1). If n is even, the group SLk (n + 1) acts transitively on the
variety of Isotk (Q) of totally isotropic subspaces of fixed dimension k. The
latter is considered as a subvariety of the Grassmann variety G(k, n + 1) of
k-dimensional subspaces of E.1 If n is odd, the latter is true if k < m. The
variety Isotm (q) of maximal isotropic subspaces has two orbits with respect
to SOk (n + 1). It consists of two connected components.
Assume n+1 = 2m and q is non-degenerate. Let U be a maximal totally
isotropic subspace of q. By Witt’s Theorem, we may choose coordinates as
in (??) such that U is given by equations t2i−1 = 0, i = 1, . . . , m. Let V
be the complementary isotropic subspace given by equations t2i = 0, i =
1, . . . , m. Then E = U ⊕ V . Let W be any other maximal totally isotropic
1
Other common notations are G(r, n + 1) if E = kn+1 or Gr−1 (n) as the set of r − 1
projective subspaces in Pn .
subspace with U ∩ W = {0} and pU , pV be the projection maps from W

to the subspaces U and V . Then the projection map pV is bijective, and
φ = pU ◦p−1 V is a linear map from V → U and the map (pU , pV ) : W → U ⊕V
has the image equal to the graph of f . Choose a basis (e1 , . . . , em ) of U
and a basis f1 , . . . , fm of V such that they correspond to the coordinates
(t1 , . . . , t2m−1 )Pin U and coordinates (t2 , . . . , t2m ) in V . Then bq (ei , fj ) = δij
and φ(f Pi ) = aij ej , i =P1, . . . , m, and W , being the graph, has a basis
(f1 − a1j ej , . . . , fm − amj ej ). Since W is isotropic, we get aii = 0
and aij = −aji . Thus, W is defined by an alternating matrix A = (aij ).
The subspace V corresponds to the zero matrix. It is clear that (λ1 , . .P . , λm )
belongs to the left nullspace of A if and only if the linear combination λi ei
belongs to V . This shows that dim W ∩V = corank A. Since the corank of an
alternating matrix is congruent to m modulo 2, we see that dim W ∩ V ≡ m
mod 2.
Let W1 , W2 be two maximal totally isotropic subspaces, suppose there
exists a third maximal totally isotropic subspace W3 such that W3 ∩ W1 =
{0}, W3 ∩W2 = {0}, then, taking U = W3 , V = W1 , we obtain that dim W1 ∩
W2 ≡ m mod 2. Conversely, suppose that this condition is satisfied. Let
us show that there exists W3 complementary to W1 and W2 . The bilinear
form bq defines an isomorphism from W1 /W1 ∩ W2 to (W2 /W1 ∩ W2 )∨ . The
condition on dim W1 ∩ W2 implies that the dimension r of these spaces is
even. Let us choose a basis er+1 , . . . , em of W1 ∩ W2 , extend it to a basis
e1 , . . . , er , er+1 , . . . , em of W1 and a basis f1 , . . . , fr , er+1 , em of W2 satisfying
bq (ei , fj ) = δij for 1 ≤ i, j ≤ r. Then define W3 to be the span of m vectors
f1 , . . . , fr , e1 + f2 , e2 − f1 , . . . , er−1 + fr , er − fr−1 . (1.1.5)
It is immediately checked that W3 a totally isotropic and is complementary

to W1 and W2 .
We say that two maximal totally isotropic subspaces W1 , W2 belong to
the same ruling if there is a maximal totally isotropic subspace W3 comple-
mentary to W1 and W2 . We have seen already that if n is odd, two subspaces
belong to the same ruling if and only if dim W1 /W1 ∩ W2 is even.
Recall that the Grassmann variety G(r, E) is a projective algebraic va-
riety. It embeds in | r E| by the Plücker map L 7→ r (L) ⊂ r (E). In
V V V
coordinates it is given by maximal minors of the matrix Z that expresses
elements of a basis of L as linear combinations of a basis in L (Plücker coor-
dinates). Since the rank of this matrix is equal to r, we can change a basis
in L to assume that one of the maximal minors is different from zero. Thus
the matrix M has m unit vectors among its columns. The rest of entries are
1.1. QUADRATIC FORMS 9
arbitrary. This defines an open subset of the Grassmann variety isomorphic

to the affine space of dimension r(n + 1 − r). The whole variety is covered
by n+1

r such open subsets. This is similar to the case when r = 1 in which
G(1, E) = |E| is the projective space. It is covered by n + 1 affine subsets
with a nonzero ith projective coordinate.
Returning to our case, where r = m and n + 1 = 2m, we obtain that
we may choose a basis in W such that the matrix M looks like [Ir A]. This
defines an affine
open subset of G(r, E) isomorphic to the affine space of
dimension n+1 r .
Proposition 1.1. Let q be an non-degenerate quadratic form.
(i) If n is even, then there is only one ruling.
(ii) If n is odd, there are two rulings.
(iii) Each ruling is an irreducible closed subvariety of the Grassmann va-

riety G(m, E) of m-dimensional subspaces in E that admits an open
cover by affine spaces of dimension m(m−1)
2 .
Proof. (i) Let W1 and W2 be two maximal totally isotropic subspace. We

choose W3 to be defined by the same basis as in (??) if r = dim W1 /W1 ∩ W2
is even. Otherwise we replace the index r with [2 2r ] and add an additional
vector 2er − fr + 2e0 , where e0 ∈ (W1 + W2 )⊥ and q(e0 ) = − 12 if p 6= 2 and
q(e0 ) = 1 otherwise.
(ii) We have seen already that two subspaces W1 and W2 are in the same
ruling if and only if dim W1 /W1 ∩ W2 is even. Two maximal totally isotropic
subspaces extending the same (m−1)-dimensional totally isotropic subspace
are obviously belong to different rulings. Thus there are two rulings.
(iii) We have seen this already in above.
We can derive the formula for the dimension of a family of maximal

isotropic subspaces in one ruling by looking at the stabilizer of one of such
space in O(E, q). We choose a standard basis defined by two complimentary
maximal totally isotropic subspaces U = he1 , . . . , em i, V = hf1 , . . . , fm i with
bq (ei , fj ) = δij. Then the matrix of bq can be written as a block-matrix
J2m = I0m I0m . A transformation σ ∈ O(EK , q) is defined by a matrix X
with entries in K/k satisfying
t
X · J2m · X = J2m .

X1 X2
If we write it in the block form X = X3 X4 , then the previous equation
is equivalent to matrix equations
t
X3 · X1 + t X1 · X3 = 0, t
X3 · X2 + t X1 · X4 = Im , t
X4 · X2 + 2 X2 · X4 = 0.
This shows that the functor K 7→ O(EK , q) is representable by an affine

algebraic variety given by the equations in entries of X (taken as unknowns)
expressing the matrix equations from above.
Also if X represents a linear operator that fixes U , then X3 = 0, and
the equations give t X1 · X4 = Im and t X3 X1 + t X1 · X3 = 0. The second
equation says that X3 is invertible, and then X4 is its inverse, and the
third equation gives m(m + 1)/2 linearly independent conditions on the
entries of X2 . It is known that the dimension of the orthogonal group
Ok (2m) is equal to m(2m − 1) (its Lie algebra is the Lie algebra of skew-
symmetric matrices of size 2m). Thus the dimension of the orbit is equal to
m(2m − 1) − (m2 + 12 m(m − 1)) = 21 m( m − 1) that agrees with the previous
computation.
Remark 1.2. Recall that a non-degenerate quadratic form q on E defines

the Clifford algebra C(q), the quotient of the tensor algebra T (E) by the
two-sided ideal generated by tensors v ⊗ v − q(v). The Clifford algebra has a
natural Z/2Z-grading C+ (q)+ ⊕ C − (q) induced by the grading of T (E). Let
ρ : E = T 1 (E) → C(q) be the natural projection map. Since it is an injective
map, we can identify E with a linear subspace of C(q). Fix a decomposition
E = U ⊕ V into the direct sum of two totally isotropic subspaces. There
is a natural isomorphism +
V∗Φ from C(q) to the algebra of endomorphisms
of the exterior algebra V uniquely determined by the properties that
Φ(u)(ω) = uxω and Φ(v)(ω)V= v ∧ ω, where u ∈ U, v ∈ V and x denote
k Vk−1
operation of contraction U ⊗ V → V , the subspace U is considered
here as the dual subspace of V via the bilinear form bq .
Let e(U, V ) be the uniqueVelement of C(q) defined by the property that
Φ(e(U, V )) is the identity on ev V and zero on odd V . Obviously, e(U, V )
V
satisfies e(U, V )2 = e(U, V ) (i.e. it is an idempotent). The center Z(q) of
C+ (q) is a 2-dimensional subspace of C+ (q) spanned by the idempotents
e(U, v) and 1 − e(V, U ). Fixing U and changing V to any W ∈ IsotU (q), we
obtain idempotents e(U, W ). Thus each W defines a decomposition of the
center (as a k-algebra) into the direct sum of two one-dimensional subalge-
bras generated by e(U, W ) and 1 − e(W, U ).
One can show that two maximal totally isotropic subspaces W, W 0 belong
to the same ruling if and only e(U, W ) = e(U, W 0 ) (see [?], Proposition 1.12).
1.2. QUADRICS 11
1.2 Quadrics
Let |E| denote the projective space of lines in E, it is equal to the projective
space P(E ∨ ) of one dimensional quotients of the dual space E ∨ . Here we
follow the notation for a projective space used, for example, in [?]. For any
extension of fields K/k, the set of K-points of |E| are lines in the linear space
EK = E ⊗k K obtained by extension of scalars. For any vector v ∈ EK we
denote by [v] the corresponding point in |E|(K). When k is algebraically
closed, we will identify |E| with |E|(k).
Recall that |E| comes with a natural line bundle denoted by O|E| (1).
Its sections are linear forms on E. Sections of O|E| (k) are elements of the
symmetric power S k (E ∨ ). In particular, a quadratic form can be considered
as a section of O|E| (2). Its zero subscheme Q is denoted by V (q). It is called
a quadric hypersurface in |E|, or just a quadric defined by equation q = 0.
We have Q(K) = {[v] ∈ |EK | : q(v) = 0}.
Proposition 1.3. A quadric Q = V (q) is smooth if and only if rad(bq ) =

{0}. The locus of singular points Sing(Q) is the projective subspace | rad(bq )|
Proof. Fix projective coordinates so that q is given by expression (??). Re-

call that the tangent space Tx X of a subvariety X of Pn at a point x ∈ X
given by homogeneous equations f1 = . . . = fN in unknowns t0 , . . . , tn+1 is
a linear projective subspace of Pn given by equations
∂fi ∂fi
(x)t1 + · · · + (x)tn+1 = 0.
∂t1 ∂tn+1
The point x is nonsingular or smooth if and only if dim Tx X = dimx X. The

set of singular points Sing(X) is equal to the set of points where dim Tx X >
dimx X.
In our casem dimx Q = n−1, so a point is singular if and only if Tx (Q) =
|E| is the whole projective space. Computing the partials, we see that
the tangent space at a point x0 = [v0 ] is defined by the linear equation
bq (v0 , x) = 0. It is the whole space if and only if v0 ∈ rad(bq ).
Note that, if v0 6∈ rad(bq ), then the restriction of the quadric to this

subspace has radical equal to hv0 i, so the corresponding quadric q|Tx0 (Q) has
x0 as its singular point. Let Ē = E/hv0 i. Then the bilinear form bq induces
a bilinear form on Ē and defines a quadric Q̄ in |Ē|. Its singular locus is
| rad(bq ) + hv0 i/hv0 i|.
A geometric interpretation of a totally isotropic subspace of q of dimen-

sion r + 1 is of course a projective subspace of dimension r (an r-plane, for
short) contained in the quadric Q = V (q). We see from the previous section
that the maximal dimension of such a subspace is equal to m = [ n−1 2 ]. A
subspace of this dimension is called a generator. Also we infer that all gen-
erators are parameterized by an algebraic variety of dimension 21 m(m − 1)
if n + 1 = 2m and consists of two connected components if n = 2m. It
is irreducible of dimension m(m + 1)/2 if n = 2m is even. The connected
component is a closed subvariety of the Grassmann variety G(m, n + 1). In
affine subsets parameterized by square matrices A of size m, the subvariety
is given by the condition that the matrix is skew-symmetric. This variety
is called the spinor variety and denoted by Sm or orthogonal Grassmannian
variety and denoted by OG(m, 2m). This variety is a homogeneous vari-
ety with respect to the spinor group Spink (n + 1), a double cover of the
orthogonal group Ok (n + 1).
By a theorem of Witt, any r-plane is contained in a ruling and all ruling
are equivalent with respect to the orthogonal group O(E, q).
Remark 1.4. It is known that the spinor group Spin(2m) admits two ir-
reducible half-spinor representations V ± of dimension 2m−1 . One can show
that the spinor variety Sm can be embedded in |V ± |. Also, one can show
that in the case n = 2m, the variety Gen(Q) of generators of a smooth
quadric in Pn is isomorphic to any irreducible component of Gen(Q0 ), where
Q0 is a smooth quadric in Pn+1 .
We can compute the dimension of the variety of generators in another

way. In fact, we compute the dimension of the variety Fk (Q) of k-dimensional
projective subspaces (for short they are called k-planes)contained in a non-
singular quadric Q in Pn for any k ≤ [ n+1 2 ]. Since this variety does not
depend on Q we denote it by Fk,n .
Consider the incidence variety
Ik,n = {(x, Λ) ∈ Q × Fk (Q) : x ∈ Λ}.
It comes with two projections p1 : Ik,n → Q and p2 : Ik,n → Fk (Q). The

fibers of the second projection are isomorphic to a k-plane, hence dim Ik =
dim Fk (Q)+k. Since a k-plane containing a point x must be contained in the
tangent space Tx Q, the fiber p−1
1 (x) is isomorphic to the variety of k-planes
on the quadric Q0 = Tx Q ∩ Q. We know that it is the cone with vertex at
x over a nonsingular quadric of dimension n − 3. Any k-plane containing x
is a cone over a (k − 1)-plane contained in Q0 . Thus p−11 (x) is isomorphic
1.3. BIRATIONAL GEOMETRY OF QUADRICS 13
to Fk−1,n−2 . This gives an inductive formula
dim Fk,n = dim Ik,n −k = dim Q+dim Fk−1,n−2 −k = n−1−k+dim Fk−1,n−2 .
It also shows that whenever Fk−1,n−2 is irreducible, then Fk,n is irreducible.

Easy computations show that
1
dim Fk,n = (k + 1)(n − k) − (k + 1)(k + 2), (1.2.1)
2
and Fk,n is irreducible if k < m = [ n2 ] or k = m when n is even.
Remark 1.5. Let ` ∈ F1 (Q) be a line on a smooth quadric. Recall the Lie
algebraVof the orthogonal group O(E, q) can be identified with the exterior
power 2 E, or, after a choice of a basis in E, with the Lie algebra of skew-
symmetric matrices of sizeVn + 1. Thus a line ` can be identified with a point
in the projective space | 2 E|. Of course, this corresponds to the Plücker
embedding of the Grassmannian G(2, E). The group G = O(E, q) has a
natural linear representation in its Lie algebra, the adjoint representation.
If we identify Lie(G) with the tangent space at G at the unity elements e,
the action is the action g 7→ (dge )−1 . Let
V us consider the corresponding
projective representation of O(E, q) in | 2 E|. The set F1 (Q) is an orbit
of this action, in fact, the only one which is closed. It is an example of
an adjoint variety which can be considered for other semi-simple algebraic
groups and their adjoint projective representations. It is also an example of
a contact Fano variety.
1.3 Birational geometry of quadrics

Choose a basis in E such that tr+1 = . . . = tn+1 = 0 are the equations of
the linear projective subspace Sing(Q). Then the equation of Q is
X
aij ti tj = 0.
1≤i≤j≤r
For any point x1 ∈ Sing(Q) with coordinates [a1 , . . . , ar , 0, . . . , 0] and a point

x2 = [0, . . . , 0, br+1 , . . . , bn+1 ] in the subspace x1 = . . . = xr = 0 the line
x1 , x2 consists of points with coordinates [αa1 , . . . , αar , βbr+1 , . . . , bn+1 bn+1 ].
Obviously it is contained in Q. This shows that Q is the join of the subspace
Sing(Q) with a nonsingular quadric Q0 in the projective subspace t1 = . . . =
tr = 0, i.e. it is equal to the union of lines joining point in Sing(Q) with
points in Q0 . In particular, when dim Sing(Q) = 0, the quadric Q is a cone

over a quadric in Pn−1 .
Recall that the product of projective subspaces |E1 | × |E2 | is a projective
variety isomorphic to the subvariety of |E1 ⊗E2 | whose points are represented
by tensors v1 ⊗ v2 , vi ∈ Ei . A closed subvariety of |E1 | × |E2 | is given by
equations representing tensors S m E1∨ ⊗ S n E2∨ . In coordinates, they are bi-
homogeneous forms of bidegree (m, n).
For example, if we take E1 = E2 = E, the diagonal ∆ in |E|×|E| is given
by the bilinear form v ∧ w = 0 expressing the condition that two vectors are
proportional.
Let φ : E1 → E2 be a nonzero linear map with kernel K, it defines a
rational map |φ| : |E1 | → |E2 | which is not defined on the subspace |K|. Its
graph is defined by
Γ|φ| := {([v1 ], [v2 ]) ∈ |E1 | × |E2 | : φ(v1 ) ∧ v2 = 0}.
It is clear that the subset of Γφ of pairs (x, y), x 6∈ |F |, is the graph of the
restriction of |φ| to |E1 | \ |F |. We have two projections
Γ|φ|
π1 π2
} "
|E1 | |E2 |.
The image of the second projection is the linear subspace |φ(E1 )| of |E2 |. Its
fiber over a point [φ(v1 )] is isomorphic to the projective subspace |K + kv1 |
of dimension dim K.
The image of the first projection is equal to |E1 |. The projection is an
isomorphism over |E1 | \ |K|. Its fiber over a point [v1 ] ∈ |K| is isomorphic to
the projective space |E2 |. If we choose a complementary subset F of φ(E1 )
in E2 , then
Γ|φ| ∼
= |F | × Bl|K| |E1 |,
where BlZ X denotes the blow-up of a variety X along the closed subvariety
Z.
We apply this construction to the case when φ is a surjective map π :
E → E/K. The map |φ| : |E| 99K |E/F | is the projection map from |E| to
|E/K| from |K|. Choosing a complementary subspace E 0 of K in E, the
projection map is the composition of the projection map to the subspace |E 0 |
and an isomorphism |E 0 | → |E/K|. The graph Γ|π| becomes isomorphic to
the blow-up Bl|K| |E| of |E| along the subspace |K|. The second projection
p1 : Bl|K| |E| → |E 0 | is a projective bundle with each fiber isomorphic to

Pdim K . The pre-image of |K| under the first projection is the exceptional
divisor Ex(π1 ) of the blow-up map π1 . It is isomorphic to |K| × |E 0 |, and
the second projection restricted to the exceptional divisor coincides with the
projection |K| × |E 0 | → |E 0 |.
Let X be a closed irreducible subvariety of |E| and p|K| : X 99K |E 0 | be
the restriction of |π| to X. If |K| ∩ X = ∅, then p|K| is a regular map. Its
image is a closed subvariety of |E 0 |. The fiber of the projection over a point
y ∈ |E 0 | is equal to the intersection of X with the subspace spanned by |K|
and y. If Z = X ∩ |K| = 6 ∅, the projection is only a rational map not defined
on Y . One can regularize it as follows.
We consider the pre-image of X \ Z in Bl|K| |E| and then take its Zariski
closure X̄ there (it is called the proper inverse transform of X in Bl|K| |E|).
The restriction of the first projection Bl|K| |E| → |E| to X̄ is a regular map
σ : X̄ → X. It is an isomorphism over X \ Z. The pre-image σ −1 (Z) is
the exceptional divisor Ex(σ) of σ. It is a closed subvariety of the exception
divisor Ex(π1 ) of the blow-up. If Z is a nonsingular subvariety of X, then
X̄ ∼
= BlZ X is the blow-up of X along Z. The exceptional divisor Ex(σ) of
σ : X̄ → X is a projective bundle over Z with fibers isomorphic to Ps , where
s = dim X − dim Y − 1. For experts, it is the projective bundle associated to
the normal sheaf NZ/X . The second projection π2 : Bl|K| |E| → |E 0 | defines
a regular map τ : X̄ → |E 0 |. Since X̄ is closed in Bl|K| |E|, the image of τ
is a closed subvariety of |E 0 |. It is called the image of the projection map
p|K| : X 99K |E 0 |.
Let us apply all of this to the case when X is a quadric Q in |E|. Choose
a projective subspace |K| of |E| and consider the projection map π|K| :
|E| 99K |E/K| with center |K|. Assume first that dim |K| = 0, i.e. |K| = x0
is a point. This is the only case when Q may not intersect the center of the
projection. Assume it does not, i.e. x0 6∈ Q. For any x ∈ Q, the line x0 , x
spanned by x0 and x intersects Q at two points counting with multiplicity.
This defines a degree 2 map Q → |E/K| ∼ = Pn−1 . Its fiber consists of two
points x1 , x2 where the line x0 , x intersects Q or one point x or one point
x if the line is tangent to Q at the point x. Let x0 = [v1 ] and x = [v]. We
have −q(v) + q(v + λv0 ) − λ2 q(v0 ) = λbq (v0 , v) for any λ ∈ k. Assume that
q(v) = 0 and bq (v0 , v) = 0, then any point y 6= x on the line x0 , x can be
written in the form y = [v + λv0 ] for some λ 6= 0. Since we assume that
q(v0 ) 6= 0, we obtain that q(v +λv0 ) 6= 0, i.e. y 6∈ Q. This shows that the line
x0 , x is tangent to Q. It is easy to see that the converse is also true. Thus
we obtain that the set Q0 of points x ∈ Q where the line x0 , x is tangent
Q is equal to the intersection of Q with the hyperplane V (bq (v0 )), where bq
is considered as a linear map E → E ∨ . This hyperplane is called the first
polar of Q with respect to the point x0 = [v0 ] and is denoted by Px0 Q.
In coordinates, it is given by the equation
n+1
X ∂q
ai = 0,
∂ti
i=1
where x0 = [a1 , . . . , an+1 ].

We have bq (v0 , v0 ) = 2q(v0 ) 6= 0, so Px0 Q 6= |E| and Q0 is a quadric in
Px0 Q 6= |E|. The map E → E/k is an isomorphism on Px0 Q and the image
of Q0 is a quadric Q̄0 in E/kv0 . Thus the projection defines a double cover
px0 : Q → |E/kv0 | ∼
= Pn−1
ramified over Q0 = Q ∩ Px0 Q and branched over the quadric Q̄0 in |E/kv0 |.
For example, assume Q has the equation
t2n+1 + l(t1 , . . . , tn )tn+1 + q(t1 , . . . , tn ) = 0,
where l isa linear form and q 0 is a quadratic form. Then the projection
from the point [0, . . . , 0, 1] is defined by the map [t1 , . . . , tn+1 ] → [t1 , . . . , tn ].
If l = 0 and p 6= 2, then the projection is a double cover branched over
the quadric V (q 0 ). If p = 2, then it is an inseparable cover if l = 0 and a
separable double cover otherwise ramified over the hyperplane V (l).
Next we consider the case when the center of the projection is a point
x0 lying on the quadric not contained in Sing(Q). In elementary geome-
try, when Q is a quadric over R and Q(R) is sphere, this is known as the
stereographic projection.
The polar hyperplane Px0 Q in this case is the tangent hyperplane of Q at
the point x0 = [v0 ]. It intersects Q along a quadric Q0 with x0 ∈ Sing(Q0 ).
As we observed before Q0 is a cone over a quadric Q00 in a projective subspace
of dimension n − 2 in |E/kv0 |. Any line joining x0 with a point x 6= x0 ∈ Q0
is contained in Q. Thus the projection blows down Q0 to the quadric Q00
lying in a hyperplane H ⊂ |E/kv0 |. On other other hand, outside Q0 , the
projection is an isomorphism onto the complement of Q00 in |E/kv0 |. The
projection map is an example of a birational map Q 99K Pn−1 . The inverse
rational map Pn−1 99K Q is given as follows. Consider the linear space of
quadratic forms q such that V (q) contains the quadric Q00 . Choose a basis
q1 , . . . , qn+1 of this linear space and consider the map given by the formulas
y 7→ [q1 (y), . . . , qn+1 (y)]. Its image is projectively isomorphic to the quadric
Q. Changing a basis we adjust it to make the image equal to Q.
For example, assume |E| = P2 with coordinates t0 , t1 , t2 . Take Q0 given
by the equations t0 = 0, t21 +t22 = 0. Then the linear space of quadratic forms
vanishing on Q0 is spanned by t20 , t0 t1 , t0 t2 , t21 + t22 Consider the rational map
P2 99K P3 defined by the formula
[t0 , t1 , t2 ] 7→ [y0 , y1 , y2 , y3 ] = [t20 , t0 t1 , t0 t2 , t21 + t2 ].
The image is a smooth quadric given by equation
y0 y3 − y12 − y22 = 0.
Note that the pre-images of hyperplanes in P3 are conics with equations

at20 + bt0 t1 + ct0 t2 + dt21 + t22 = 0. Over reals they are circles.
sectionCohomology of quadrics Since a quadric Q ⊂ |E| is a hypersurface
in Pn , we can apply computations of cohomology from [Hartshorne]. We
have
H i (Q, OQ (m)) = 0, i 6= 0, n, m ∈ Z;
H 0 (Q, OQ (1)) = n + 1, H 0 (Q, OQ (m)) ∼
= S m (E ∨ )/QS m−2 (E ∨ ), m ≥ 2;
H n (Q, OQ (m)) ∼= H 0 (Q, OQ (−n + 1 − m),
The last isomorphism uses that the canonical sheaf ωQ of a quadric is iso-
morphic to OQ (−n + 1).
We can also compute topological cohomology. We use the l-adic coho-
mology or usual cohomology if k = C.
Proposition 1.6. Let Q be a smooth quadric in Pn and ηQ = c1 (OQ (1) be
the divisor class of its hyperplane section. Then
(i) Pic(Q) ∼
= Z if n 6= 3 and Pic(Q) ∼
= Z2 if n = 3.
(ii) The Chern class homomorphism c1 : Pic(X) → H 2 (X, Zl ) is an iso-
morphism.
(iii) H 2i+1 (Q, Zl ) = 0;
(iv) H 2k (Q, Zl ) is freely generated by ηQ

k if n is even, or n is odd and
n = 2k + 1.
(v) H 2k (Q, Zl ) ∼ k = [r ] + [r ]
= Z` [r1 ] + Z` [r2 ] if n = 2k + 1, where ηQ 1 2
and r1 , r2 are the cohomology classes of generators from two different
rulings of Q.
Proof. We use induction by n. Let X → Q be the blow-up of a point x0 ∈ Q.

We know that X is isomorphic to the blow-up of a smooth quadric Q0 in a
hyperplane H ⊂ Pn−1 . The assertion will follow from the known behaviour
of cohomology under a blow-up f : X → Y of a smooth closed subvariety Z
in Y . We have
H ∗ (X, Zl ) ∼
= f ∗ H ∗ (X, Zl ) ⊕ H ∗ (E, Zl )/g ∗ H ∗ (Z, Zl ),
where is the exceptional divisor and g : E → Z is the restriction of f to

E (see [Griffiths-Harris], p. 605 when k = C). Also, we have to use that
E = P(NZ/Y ) is the projective bundle over Z associated to the normal sheaf
NZ/Y ∼= (IZ /IZ2 )∨ , where IZ is the ideal sheaf of Z in Y . It is known
that H ∗ (E, Zl ) is a free module over H ∗ (Z, Zl ), where the structure of a
module is defined by the homomorphism g ∗ . Its generators are ηE/Zi , where
ηE/Z = c1 (OE (1)). they satisfy a relation
r r−1
ηE/Z + c1 ηE/Z + · · · + cr = 0,
where ci = ci (NZ/Y ) are the Chern classes of NZ/Y and r = codim(Z, Y ) =

rank(NZ/E ). We leave to the reader to finish the proof by computing coho-
mology of X in two ways, as the blow-up of Q0 in Pn−1 and as the blow-up
of a point on Q.
1.5. Examples.
Example 1.7. A one-dimensional quadric Q = V (q) is called a conic. It is

singular if and only if the rank of q is less than 3, hence if and only if it is
irreducible over an extension of k. A conic with a k-point is isomorphic to
the projective line via the projection from this point. A conic of rank 1 is a
double line.
The polar line of a conic with respect to a point p0 outside of the conic
intersects the conic at two different points x, x0 . The tangent lines at these
points intersect at p0 .
All conics in |E| are parameterized by the 5-dimensional projective space
|S 2 E ∨ | ∼
= P5 . Singular conics are parameterized by the discriminant hyper-
surface D2 . It is a cubic hypersurface in P5 . In coordinates, its equation is
given by the determinant of a symmetric 3 × 3-matrix det(Aij ). Its singular
locus parametrize double lines. It is isomorphic to the Veronese surface in
P5 , the image of the Veronese map ν2 : P2 = P(E ∨ ) → P5 = P(S 2 E ∨ ) given
by [l] → [l2 ].
Example 1.8. A quadric in P3 is a quadric surface. A smooth quadric

surface over an algebraically closed field k can be given by equation
t1 t2 + t3 t4 = 0.
It has two families of lines given by equations
λt1 − µt3 = µt2 + λt4 = 0,

λt2 − µt4 = µt1 + λt3 = 0.
The lines of each family do not intersect, each line from family intersects all
lines in another family. The Segre map
P1 × P1 → Q, ([u0 , u1 ], [v0 , v1 ]) 7→ [u0 v0 , u0 v1 , u1 v0 , u1 v1 ]
is an isomorphism. The images of fibers of each projection P1 × P1 → P1 is

a family of lines on Q.
The group of automorphism of a smooth quadric contains a subgroup of
index 2 isomorphic to Aut(P1 ) × Aut(P1 ). It is also isomorphic to PSLk (4).
The quotient by this subgroup is generated by the involution that switches
the factors. Thus, we have
Aut(Q) ∼
= (PGLk (2) × PGLk (2)) o Z/2Z.
The projection from a point x0 ∈ Q to P2 extends to a regular map from

the blow-up Blx0 Q to P2 .
Blx0 Q
σ π
| #
Q / P2 .
The map σ is an isomorphism outside two points y1 , y2 in P2 . The fibers over

these points are the proper transforms of two generators of Q containing the
point x0 . The inverse map P2 99K Q given by conics through the points
y1 , y2 . It decomposes as in the following diagram:
Bly1 ,y2 P2
σ π
{ #
P2 / Q.
The two families of lines are the images of pencils of lines through the points
y1 , y2 under the inverse rational map P2 99K Q. The image of the line y1 , y2
is the point x0 .
A quadric of corank 1 has an isolated singular point. It is called a
quadratic cone because it is isomorphic to the cone over a conic. A quadric
of corank 2 is the union of two planes. Its singular locus is the intersection
of the planes. A quadric of corank 1 is the double plane. It is singular
everywhere.
The space of quadrics |S 2 E ∨ | is isomorphic to P9 . Singular quadrics form
the discriminant hypersurface D3 . It is a quartic hypersurface in P9 given
by the discriminant equation. Its singular locus parameterizes quadrics of
corank ≥ 2. It is isomorphic to the quotient of P2 × P2 by the involution
that switches the factors. It is the image of the map P2 × P2 → P9 given by
the linear system of |S 2 E ∨ | ⊂ |E ∨ ⊗ E ∨ |. Its singular locus is the image of
the diagonal, and isomorphic to P2 .
Example 1.9. Let n = 4, a nonsingular quadric over an algebraically closed
field k can be given by equation
t1 t2 + t3 t4 + t25 = 0.
The projection map πx0 : Q 99K P3 decomposes according to the following
commutative diagram
Blx0 Q
σ π
| #
Q P3 .
We know that the inverse map of the projection map Q 99K P3 from a
point x0 ∈ Q is given by quadrics containing some nonsingular conic C in
a hyperplane H ⊂ P3 , the image of the tangent hyperplane Tx0 Q. We have
an irreducible 2-dimensional family of lines on Q. Each line containing x0
is blown down to a point in P3 . All the images of such lines lie in the conic
C. Any other line intersects Tx0 Q, and hence intersects some line through
x0 . Its projection is a line in P3 that intersects a conic C at one point.
Conversely, the inverse map P3 99K Q is given by quadrics containing the
conic C. It decomposes as in the following diagram:
BlC P3
σ π
| "
P3 / Q.
1.4. APPENDIX 1:QUADRICS IN CHARACTERISTIC 2 21
Each line intersecting C at two points is blown down to the point x0 . The
family of lines on Q is the image of the family of lines in P3 that intersect
C at one point.
The set of lines F1 (Q) is an irreducible 3-dimensional variety isomorphic
to P3 .
Example 1.10. Assume n = 5. a nonsingular quadric in P5 over an alge-
braically closed field k can be given by equation
t1 t2 + t3 t4 + t5 t6 = 0.
Consider P5 as the projective space | 2 U |, where U is a 4-dimensional
V
linear space. VLet e1 , e2 , e3 , e4 be a basis in U , then ei ∧ ej , 1 ≤ i < j ≤ 4
is a basis in 2 . For any two distinct points [v], [w] in |U |, V the line joining
these two points defines a decomposable 2-vector v ∧ w in 2 U . Obviously,
each decomposable 2-vectors is obtained from a unique line in P3 . It is easy
to check that a 2-vector λ is decomposable if and only if λ ∧ λ = 0. V Thus,
the Grassmannian of lines G(2, U ) is isomorphic to the subvariety of || 2 U |
defined by the condition that λ ∧ λ = 0. In coordinates pij dual to the basis
(ei ∧ ej ) (they are the Plücker coordinates), the condition is translated into
the condition
p12 p34 + p13 p24 − p14 p23 = 0.
Thus we see that any nonsingular quadric Q in P5 is isomorphic the Grass-
mannian G(2, 4) of lines in P3 . A For this reason, the Grassmannian G(2, 4)
is called sometimes the Klein quadric. Now, we easily find two irreducible
3-dimensional families planes in Q. One family consists of planes σy formed
by lines in P3 passing through a point y ∈ P3 . Another family consists of
planes σΛ formed by lines in P3 contained in a plane Λ ⊂ P3 . A line in Q is
formed by lines in P3 that are contained in a plane Λ and contains a fixed
point y ∈ Λ. Obviously, the family of lines is of dimension 5. It agrees with
formula (??).
We have PO(Q)0 ∼ = Aut(P3 ) ∼
= PGL(4). The group Aut(P3 ) acts on Q
via its identification with the Grassmannian G(2, U ). An extra automor-
phism comes from a polarity which is an isomorphism |U | → |U ∨ | defined
by a linear map c : E → E ∨ such that c ◦ t c−1 is the identity (see [Classical
Algebraic Geometry], p. 10).
1.4 Appendix 1:Quadrics in characteristic 2

The first different feature of this case is that a quadratic form is not deter-
mined by its polar bilinear form. In fact we see that bq is an alternating
P 2
form in this case, and, for example, the quadratic form ti has zero polar
bilinear form.
A vector in the radical of q may not be in the zero set of q, so a singular
vector in this case means that it is an element v ∈ Rad(bq ) such that q(v) =
0. The jacobian criterion of smoothness implies that a singular point of the
associated quadric Q = V (q) is equal to [v], where v is a singular vector.
Since Q(k̄) either contains |Rad(bq )|(k) or intersects it along the locus V (l2 ),
where l is a linear form, we see that the singular locus Sing(Q) of Q is a
linear subspace of dimension equal to δ − 1, or δ − 2, where δ = dim Rad(bq )
is the defect of q. In particular, if n is odd, the defect of bq must be even, and
hence Q is singular if and only if bq is degenerate. On the other hand, if n is
even, the defect is odd, and Q could be smooth but bq is always degenerate.
If bq is degenerate the determinant of its matrix is zero, but this does
not imply that V (q) is singular. The notion of the discriminant has to be
replaced with the notion of the 21 -discriminant.
Exercise 1.11. Let B be a skew-symmetric matrix of odd size n + 1. Let

Pf 1 , . . . , Pf n+1 are the pfaffians of the principal submatrices of B obtained
by deleting the ith row and the ith column of B.
(i) Show that V (q) is smooth if and only
q(Pf 1 , . . . , Pf n+1 ) = 0
has no solutions in any extension of k.
(b) Show that the left-hand side is a polynomial of degree n + 1 in the

coefficients aij of q. It is called the 21 -discriminant of q (Of course the
discriminant of any q is the determinant det(B) which is zero if n + 1
is odd and char(k) = 2).
(c) Compute the 12 -discriminant for small n and guess the general form of
it (it must be analogous of the general form of the discriminant).
P
(d) Let q = Aij ti tj be a quadratic form whose coefficients are indepen-
dent variables. The determinant of the matrix of the polar bilinear
form is a polynomial D in Z[(Aij ). Show that D = 2D0 , for some
polynomial D0 ∈ Z[(Aij ]).
(e) Show that, for any homomorphism φ P : Z[(Aij ) → k, the image φ(D0 )
0 1
of D is the 2 -discriminant of φ(q) = aij ti tj , where aij = φ(Aij .
1.5. APPENDIX 2:ORTHOGONAL GRASSMANNIANS 23
(f ) Show that when n = 2 and n = 4, the equations of 21 -discriminant for

conics and quadrics of dimension 3 are
Disc02 = A11 A223 + A22 A213 + A33 A212 + A12 A23 A13 ,
Disc04 = (A11 A223 A245 + · · · ) + (A212 A34 A45 A35 + · · · )
+(A12 A23 A34 A45 A15 + · · · )
We say that a subspace F is totally singular if the restriction of q to F is

zero. In geometric terms this means that |F | is a linear projective subspace
contained on Q. A totally singular subspace is totally isotropic (i.e the
restriction of bq to F is zero) but the converse is not true. In coordinates,
P 2
the restriction of q to a totally isotropic subspace F of bq is equal to ai ti ,
∨
where ti is a basis in F . Thus, over the algebraic closure k̄ of k, it is equal
to the square of a linear function l : F → k. In particular, the restriction of
q to F is either the whole F or a hyperplane F0 in F . In geometric terms,
|F | or F0 | is a linear projective subspace contained in Q(k̄). It follows from
the equations (??) that a maximal isotropic subspace of Q coincides with a
maximal singular subspace if n + 1 = 2m and its dimension is larger by one
if n = 2m. So, we have the same properties of families of generators as in
characteristic p 6= 2.
The restriction of q to | rad(bq )| is either the whole space or a hyperplane.
This shows that the set of singular vectors of q is a linear subspace in Ekperf
of dimension δ(q) or δ(q) − 1. The set of the corresponding points in the
quadric Q = V (q) is the locus Sing(Q) of singular points in Q.
Assume p = 2. If v0 ∈ rad(bq ), then Px0 Q = |E| and any line x0 , x, x ∈ Q
is tangent to Q at x. The projection px0 is an inseparable map of degree
2 onto Pn−1 . The point x0 is called the strange point of the quadric Q.
For example, when n is even, there is always such point. If n is odd such
point exists only if bq is degenerate. If v0 6= rad(bq ), then Px0 Q 6= |E| but
x0 ∈ Px0 Q. The first polar is tangent to Q at any intersection point with
Q. Thus Q0 becomes the double hyperplane of dimension n − 2 contained
in Q. The projection is a separable double cover ramified along Q0 . The
image of Q0 in |E/kv0 | is a hyperplane. This is the branch divisor of the
double cover.
1.5 Appendix 2:Orthogonal Grassmannians

Let Q = V (q) be a non-degenerate quadric in an odd-dimensional projective
space P2k+1 = |E|. In this Lecture we have introduced the orthogonal
Grassmannian OG(k + 1, 2k + 2), a connected component of the variety

of linear subspaces of dimension k contained in Q. Its dimension is equal
to 21 k(k + 1). Via the Plücker embedding, the orthogonal Grassmannian
2k+2
embeds in | 2 E| =
V ∼ P( k+1 )−1 . However, the image is contained in a proper
projective subspace |S| of dimension 2k−1 . The linear subspace S of E is
one of two irreducible representations of the orthogonal group O(E, q) of
dimension 2k , the half-spinor representations.
Let us recall its definition. Let C(q) be the Clifford algebra of (E, q). It is
the quotient of the tensor algebra T ∗ (E) by the two-sided ideal generated by
elements v⊗v−q(v). The natural Z/2Z-grading of the tensor algebra induces
a natural grading C(q) = C(q)+ ⊕ C(q)− of the Clifford algebra. Suppose
q admits a decomposition E = F ⊕ F 0 into the direct sum of to totally
isotropic subspaces. It always happens if the ground field is algebraically
closed, and, we make this assumption. Using the restriction of the bilinear
form bq : E → E ∨ to F 0 ×F , we can naturally identify F 0 with The dual linear
space F ∨ . The subalgebra of C(q) generated byV vectors f ∈ F ⊂ E = T 1 (E)
is naturally isomorphic to the exterior algebra • (F ) of dimension 2k+1 . It
has aVnatural V Z/2Z-grading and decomposes accordingly. into the direct
sum • (F ) = •+ (F ) ⊕ •− (F ). Each summand is linear space of dimension
V
2k . Let us define a homomorphism of algebras

•
^
ρ : C(q) → Endk ( (F )).
−
V•
Let f ∈ F (resp. f ∗ ∈ F ∨ ). It acts on (F ) via the wedge-product (resp.
the contraction). This action first extends, by additivity to the whole E,
and then by the derivation rule to the tensor algebra. It is checked that the
elements of the two-sided ideal act as the zero operators. Each summand
is obviously invariant. InVthis way one identifies C(q) with the algebra of
endomorphisms of Endk ( • (F )) isomorphic to the algebra of matrices of
size 2k+1 . The subalgebra C(q)+ is isomorphic to the sum of two left ideals
in the matrix algebra, each is isomorphic to the algebra of matrices of size
2k . These are two half-spinor representations S± of C + (q).
Let G(q) be the group of invertible elements x in C + (q) such that, for
any v ∈ E, one has x · v · x−1 ∈ E. The image of the homomorphism
φ : G(q) → GL(E) is obviously contained in the orthogonal group O(E, q).
Suppose char(k) 6= 2. It is known that the group O(E, q) is generated by
reflections sr : v → v − (v, r)r, where r ∈ E, q(e) = 1. Since r ⊗ r = 1 in
T (E), we obtain that r2 = 1 is C(q), hence r ∈ G(q). It is immediately
checked that φ(r) = −sr and, conversely, the image of any element r in
G(q) ∩ E is equal to −sr . Using this one can show that homomorphism φ is
surjective (we have to use that the element − idE in O(E, q) is the product
of even number of reflections). The kernel of the homomorphism φ is equal
to the group Z ∗ of invertible elements in the center of the algebra C + (q).
The extension
1 → Z ∗ → G(q) → O(E, q) → 1
does not split, however, there is a unique subgroup Spin(q) in G(q) that is
mapped surjectively onto O(E, q) with kernel {
pm1}. The group Spin(q) is called the spinor group of (E, q). Being the
subgroup of G(q) it admits two linear representations S± , called the half-
spinor representations of Spin(q). They are isomorphic (via the exterior
automorphism of the group) irreducible representations of dimension 2k .
Being a closed subvariety of the usual GrassmannianVG(k +1, 2k +2), the
orthogonal Grassmannian embeds in the Plücker space | k+1 (E)|. However,
it spans a proper projective subspace of the Plücker space. let us explain
which one.
Let bq : E → E ∨ be the linear isomorphism define by the non-degenerate
quadratic form q. By passing to theVexterior powers,
k+1 m Vm for∨ any m ≥
Vm ∨it defines,
0, a linear isomorphism ∧ (bq ) : (E) → (E ) = (E) . Taking
m =Vdim E, we can identify the one-dimensional linear spaces 2k+2 (E ∨ )
V
and 2k+2 (E ∨ ) with the field of scalars k (by assuming that the discriminant
of q is equal to 1). The wedge-product defines a non-degenerate pairing
k+1
^ k+1
^ 2k+2
^
(E) × (E) → (E) = k
: V k+1 (E ∨ ) → k+1 (E ∨ ). The

V V
and, hence defines a linear isomorphism
automorphism ∧k+1 (bq ) ◦ −1 : k+1 (E) → k+1 (E) is an involution with
V
two eigenvaluesV±1 that decomposes k+1 (E) into the direct sum of two
V
eigensubspaces k+1 (E)± of the same dimension. Under the Plücker space,
the image of each connected component Fk (Q)V ± of the variety Fk (Q) is equal
to the intersection of G(k + 1, 2k + 2) with | k+1 (E)± |. Thus we see that
the orthogonal Grassmannian is a linear section of the usual Grassmannian.
Example 1.12. Let Q be a non-degenerate quadric in |E| = P3 , so that

k = 1 and F1 (Q)± parameterizes lines in one of the two V rulings of lines in
Q. We know that G(2, 4) is the Klein quadric in P5 = | 2 E|. The image
of F1 (Q)± is a conic in G(2, 4) cut our by a plane in P5 .
If we take Q to be a quadric in |E| = P5 , for example the Klein quadric
G(2, 4) = G1 (P3 ) in P5 , then we know that F2 (Q)± are the two families of
planes (of lines passing through a point in P3 or lines contained in a plane

in P3 ). V
They are naturally isomorphic to P3 (or the dual P3 ). In the Plücker
space | 3 E| ∼= P19 they are equal to the intersection G(3, 6) = G2 (P5 ) with
P . One can show that the embedding P3 ,→ P9 obtained in this way is the
9
second Veronese embedding ν2 : P3 → P9 .
V It follows from the previous example that the embedding O(k + 1, E) in

| k+1 (E)± | is not a minimal projective embedding. The minimal embed-
ding is the spinor embedding. which we proceed to describe. Fix a maximal
totally isotropic subspaceV F of E and consider the associated half-spinor rep-
• Vk+1
resentation S± based on (E)± . The line (F ) is spanned by a vector
belonging to the subspace S+ if k is odd and to the subspace S− otherwise.
Consider the orbit of Spin(E, q) of this line in its natural action in the pro-
jective space |S± |. Of course, the projective representation of Spin(E, q) in
|S± | defines a projective representation of the orthogonal group O(E, q) but
it does not leave to a linear representation of this group (this why do we
need the spinor group!). Since O(E, q) acts transitively on totally isotropic
subspaces of the same dimension, the orbit coincides with Fk (Q)| pm|. One
can show that the orbit spans S± . In fact, the orbit coincides with a ho-
mogeneous space of the form Spin(E, q)/P , where P is a maximal parabolic
subgroup of an algebraic group Spin(E, q) with Spin(E, q)(k) = Spin(E, q).
Its Picard group is generated by a line bundle that defines the embedding
Fk (Q) into |S± .
Example 1.13. In the previous example, we have seen the spinor embed-
dings of O(2, 4) and O(3, 6) into P1 and P3 , respectively. The 6-dimensional
variety O(4, 8) representing one family of 3-spaces in a nonsingular quadric
Q ⊂ P7 is minimally embedded in P7 . It is a nonsingular quadric in P7 . In
fact, it must a hypersurface in P7 , the only homogeneous hypersurface in a
projective space is either a nonsingular quadric or a projective subspace.
Example
V• 1.14. The 10-dimensional variety O(5, 10) embeds into P15 =
| ± (F ). One can show see [Kapustka, Projections of Mukai varieties] that
it is given as the zero locus of the quadratic map
•
^ 2
^ 4
^ 4
^
Φ : (F ) = k ⊕ F⊕ F → F ⊕F
±
defines as follows. We first identify k with 5 F and 4 F with F ∨ . Then

V V
we set
1
Φ(v, ω, φ) = (vxφ + ω ∧ ω, vxω).
2
The zero locus is given by 10 quadratic equations.

The homogeneous variety O(5, 10) is an example of a Muklai variety, a
non-degenerate subvariety of a projective space whose general linear sections
of dimension one are canonically embedded algebraic curves of some genus
g > 1. In particular, the degree of such a variety is equal to 2g − 2. In our
case, g = 7 and the curve is cut out by a general projective subspace in P15
of codimension 9. A linear section by a subspace of codimension 8 (resp. 7)
is a Fano 3-fold (resp. a K3 surface) of degree 12.
Lecture 2
Normal elliptic curves
2.1 First example

We start with the intersection of two irreducible conics C1 ∩ C2 . For sim-
plicity, let us assume that the field is algebraically closed. By Bezout’s
theorem, C1 ∩ C2 consists of k ≤ 4 points. Let C1 = V (q1 ), C2 = V (q2 ) and
C(t1 , t2 ) = V (t1 q1 + t2 q2 ). When [t1 , t2 ] runs P1 , we have a pencil of conics.
Let D be the discriminant of the conic t1 q1 + t2 q2 . It is a homogeneous
polynomial a1 t31 + a2 t21 t2 + a3 t1 t22 + a4 t32 of degree 3 in t1 , t2 . Plugging in
(t1 , t2 ) = (1, 0) and (0, 1), and using that C1 , C2 are smooth, we get that
a1 , a4 6= 0. Clearly, V (D) consists of r ≤ 3 points.
The number r is equal to the number of singular conics in the pencil. If
two conics intersect at a point P with multiplicity m, then there exists a
member in the pencil that has a point of multiplicity ≥ m at this point. If
k = 4, and C1 ∩ C2 = {p1 , p2 , p3 , p4 }, then we have three singular members
formed by three pairs of lines hpi , pj i + hpk , pl i. In fact, no three points are
collinear, because otherwise all conics in the pencil contain a line as their
irreducible component. So, we may choose projective coordinates such that
the points are [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]. Then the conics in the pencil
have equations
at0 t1 + bt0 t2 + ct1 t2 = 0, a + b = c = 0.
The three singular conics are V (t0 (t1 − t2 )), V (t1 (t2 − t3 )) and V (t2 (t0 − t2 )).
If k = 3 and C1 ∩C2 = 2p1 +p2 +p3 , then we have two singular members.
One is the union of the line hp2 , p3 i and the common tangent line to the
conics at the point p1 . The second one is the union of the lines hp1 , p2 i and
hp1 , p3 i.
29
30 LECTURE 2. NORMAL ELLIPTIC CURVES
If C1 ∩ C2 = 2p1 + 2p2 , then we have a singular member equal to the

union of tangent lines at p1 and p2 and the double line hp1 , p2 i.
If C1 ∩ C2 = 3p1 + p2 , we have one singular member, the union of the
tangent line at p1 and the line hp1 , p2 i.
Finally, if C1 ∩C2 = 4p, we have one singular member, the double tangent
line at p.
So, we see that the number r depends on k in a rather complicated way. If
p 6= 2, the different cases are determined by considering elementary divisors
of the λ-matrix det(λA + B), where A, B are matrices of the bilinear forms
bq1 , bq2 . Possible Jordan forms are the following: one with three distinct
eigenvalues (r = 3), two with two distinct eigenvalues (r = 2), and two with
one eigenvalue (r = 1). We will discuss this later.
Let f : P2 99K P1 be the rational map defined by
x = [x0 , x1 , x2 ] 7→ [q1 (x), q2 (x)]. (2.1.1)
Let X be the Zariski closure of the graph of the map f 0 : P2 \ C1 ∩ C2 → P1 .

If k = 4, X is a nonsingular surface, and the projection to P2 blows up the
intersection points. It is a del Pezzo surface of degree 5.
Recall from [Dolgachev,Classical Algebraic Geometry] that a del Pezzo
surface is a smooth algebraic surface S such that −KS is ample. Each such
surface is a rational surface. It is isomorphic to P2 or P1 × P1 , or admits a
birational morphism f : S → P2 whose inverse is the blow-up of k points
p1 , . . . , pk in the plane. Since KS2 > 0, the number k is at most 8. We have
KS2 = 9 − k (under a blow-up the self-intersection of the canonical class
decreases by one). The number d = KS2 is called the degree of the del Pezzo
surface S. For d ≥ 3, the linear system | − KS | defines an embedding of
S into Pd as a surface S ac of degree d. It is called an anti-canonical model
of a del Pezzo surface. Conversely, every smooth surface of degree d in Pd
not lying in a proper projective linear subspace is isomorphic to a del Pezzo
surface of degree d. The linear system | − KS | defines a birational map
P2 99K S ac onto an anti-canonical model. By adjunction formula, for any
smooth curve C, we have C 2 + C · KS = 2g(C) − 2, where g is the genus of
C. In particular, if C is a (−1)-curve, i.e. a smooth rational curve C with
self-intersection C 2 = −1, we have (−KS ) · C = 1. Thus its image on S ac is
a line. A (−1)-curve is characterized by the property that it can be realized
as the exceptional curve of the blow-up of a nonsingular point on a surface.
Or, equivalently, it can be blown down to such a point. Under the birational
morphism S → P2 , the image of a (−1)-curve R is an irreducible curve of
some degree m. Since under the blow-up the self-intersection of the proper
2.1. FIRST EXAMPLE 31
inverse transform of an irreducible curve R̄ decreases by one, we see that

2
P
m − a = −1, where a = mi with mi equal to the multiplicity of R̄ at the
point pi . This implies that for d ≥ 3, the curve R is either blown down to
one of the points pi , or m = 1 and a = 2, or m = 2 and a = 5. The linear
system | − KS | is defined by plane cubic curves passing through the points
p1 , . . . , p5 . If S = P2 , then | − KS | is the complete linear system of plane
cubics and it defines a Veronese map ν3 : P2 → P9 onto a Veronese surface
of degree 9 in P9 . An anti-canonical del Pezzo surface S ac of degree d ≥ 3
is isomorphic to a projection of a Veronese surface of degree 9 into a linear
subspace of dimension d ≥ 3.
Let us go back to our case when d = 5. We denote a del Pezzo surface
of degree 5 by D5 . It is unique, up to isomorphism. By above a (−1)-
curve on D5 is either one of four exceptional curves or equal to the proper
inverse transform of one of the six lines hpi , pj i. Altogether with have ten
(−1)-curves on D5 . Their images in the anti-canonical model S ac are lines.
It is known that the canonical sheaf of the Grassmannian G = G(k, m)
is isomorphic to OG (−m). In particular, for the Grassmannian G(2, 5),
we have ωG ∼ = OG (−5). By the adjunction formula, if H is a hyperplane
section of G, we have ωH ∼ = OF (−4). Continuing in this way, we see that
the intersection of G)2, 5) by a linear subspace of codimension 4 is a surface
X in P5 with canonical sheaf isomorphic to OX (−1). It is known that the
degree of G(2, 5) in its Plücker embedding is equal to 5, so X is of degree
5 and, hence must be isomorphic to an anti-canonical model S ac of a del
Pezzo surface of degree 5.
We know that a del Pezzo surface of degree 5 is unique up to isomor-
phism. So that the freedom of choosing a linear subspace of codimension
4 is illusory. Different choices lead to projectively isomorphic surfaces. To
see this we count parameters. Codimension 4 linear subspaces in P9 , where
G(2, 5) is embedded, depend on dim G(6, 10) = 24 parameters. On the other
hand, the projective linear group PGL(5) acts in P9 via its natural action
in P4 . Its dimension is equal to 25 − 1 = 24. Thus we expect that all
codimension 4 sections of G(2, 5) form one orbit with respect PGL(5).
One
Vm more remark is in order. A Grassmann variety G(m, n+1) embedded
in | (k n+1 )| via the P”ucker embedding is either isomorphic to Pn (when
m = n or m = 1) or it is equal to the intersection of quadrics. The quadrics
are given by the Plücker quadratic relations between the Plücker coordinates.
m+1
X
(−1)k pi1 ,...,im−1 ,jk pj1 ,...,jk−1 ,jk+1 ,...,jm+1 = 0, (2.1.2)
k=1
where (i1 , . . . , im−1 ) and (j1 , . . . , jm+1 ) are two strictly increasing subsets
of [1, n + 1]. These relations are easily obtained by considering the left-
hand-side expression as an alternating (m + 1)-multilinear function on Cm .
It is known that these equations define G(m, n + 1) scheme-theoretically in
n+1
P( m )−1 (see, for example, [Hodge-Pedoe, vol. 2].
Consider the rational map φ : P2 99K P1 (??) and let
D5
π f
~ φ
P2 / P1
be its resolution of indeterminacy points. The map π is the blow-up of

the points p1 , p2 , p3 , p4 and the map f is a conic bundle. The latter means
that its general fiber is isomorphic via the projection π to an irreducible
conic. Its singular fibers are isomorphic to reducible conics. In our case, we
have three such singular fibers. They are the reducible conics in the pencil
P = V (λq1 + µq2 ) of conics. The irreducible components of the fibers are
the proper transforms of lines hpi , pj i. Let us denote them by Uij . The
exceptional curves over the points pi we denote by Ui5 . Thus we have 10
(−1)-curves Uab , 1 ≤ a < b ≤ 5. In the anti-canonical model S ac of S they
are 10 lines. It is easy to see that each curve Uab intersects three other
curves. If we define the graph with vertices Uab and draw the edges when
two curves intersect, we obtain the famous Petersen graph:
Figure 2.1: Petersen graph
The pair (D5 , f ) has a nice moduli-theoretical interpretation. Take a

2.2. MODULAR SURFACES 33
point x ∈ S which does not lie on the union U of the curves Uab . Then
the fiber C(x) over f (x) is an irreducible conic on which four ordered points
are fixed, they are the intersections of the curves Ri5 with this fiber. Or, in
other words, they are the points which are mapped to the points p1 , . . . , p4
under the morphism π. Since C(x) is isomorphic to P1 , we get an ordered
set of 4 points on P1 . Choosing coordinates in P1 , we may assume that the
reducible fibers correspond to the points {0, 1, ∞}. In this way, P1 \ {0, 1∞}
becomes isomorphic to the moduli space M0,4 of ordered sets of 4-points in
P1 modulo projective equivalence. The pre-image of this open subset of P1 is
the open set D5o = D5 \ U. A point x in this open set defines a fiber through
this point identified with P1 and an ordered set of 5 points (p1 , p2 , p3 , p4 , x).
In this way S o becomes isomorphic to the moduli space M0,5 of rational
curves together with an ordered set of 5 points on it. The whole curve P1
(resp. the surface D5 ) becomes isomorphic to a compactification M0,4 (resp.
M0,5 ).
2.2 Modular surfaces

If p 6= 2, the conic bundle f : S → P1 has also a natural moduli interpretation
as the Kummer universal family for the fine moduli space of elliptic curves
with a 2-level structure.
Let us explain what these words mean. First, a N -level structure on an
elliptic curve C is a choice of a basis in the group E[N ] of its N -torsion
points. We assume that (N, p) = 1. If N = 2, the negation involution
τ : x 7→ −x has the quotient C/(τ ) isomorphic to P1 . The double cover π :
C → P1 ramifies at 4 points, the 2-torsion points. If we use the coordinates
on P1 such that the zero point p0 goes to ∞, the complement of p0 has the
Weierstrass affine equation
f (x, y) = y 2 + x3 + a2 x2 + a4 x + a6 = 0
The point (x, y) = (xi , 0), i = 1, 2, 3 on the curve are the non-trivial 2-
torsion points. An order on the set of these points defines a basis in the
group E[2]. So, we see that an elliptic curve with level 2 structure is defined
by an ordered set of 4 points on P1 . Thus, we can identify M0,4 with the
moduli space of elliptic curves with level 2-structure. It is isomorphic to
P1 \ {0, 1, ∞}.
The fine moduli space of elliptic curves with N -level structure exists and
is isomorphic to an affine curve X(N )o . Over C this curve is isomorphic
to the quotient of the upper half-plane H = {z = a + bi ∈ C : b > 0} by
the modular group Γ(N ) of matrices from SL(2, Z) congruent to the identity
matrix mod N . It admits a compactification X(N ), called the modular curve
of level N . The complement consists of
1 Y 1
cN := N 2 (1 − 2 )
2 p
p|N
points, called cusps.. For N ≥ 3, the moduli space is a fine moduli space
and the universal curve
π : S(N ) → X(N )
over the moduli space exists. It is a modular elliptic surface of level N . For
N = 2, the moduli space exists as a stack only.
Recall that an elliptic surface is a smooth projective algebraic surface
S together with a morphism f : S → B (called an elliptic fibration) to a
smooth projective curve B such that all fibers over an open non-empty subset
are curves of genus one. An elliptic surface is called relatively minimal if
any birational morphism S → S 0 over B is an isomorphism. It follows from
the theory of algebraic surfaces that this is equivalent to that no fiber of
f contains a (−1)-curve. All possible singular fibers of relatively minimal
elliptic surfaces, considered as positive divisors, were described by Kodaira.
They are distinguished by Kodaira’s types II, III, IV, In , In∗ , II ∗ , III ∗ , IV ∗ .
Irreducible fibers have one ordinary double point (type I1 ) or one ordinary
cusps (type II). Other fibers consist of n components if type is In , n + 6
components of type is In∗ , two components if type III, three components if
type IV, n = 7, 8, 9 if the type is II ∗ , III ∗ , IV ∗ . The intersection of these
components is given by the following graphs:
Here we use another notation that comes from the theory of root systems
of simple Lie algebras. The graphs which you see are Dynkin diagrams of
affine root systems of types Ãn , D̃n , Ẽ6 , Ẽ7 , Ẽ8 .
Let Eη be the fiber of f : S → B over the generic point η of B (in the
scheme-theoretical sense). This is a curve over the field of rational functions
k(η) of the curve B. It has a rational point over this field if and only if
the fibration f admits a section s : B → S. We identify this section with
its image, a smooth curve isomorphic to B under the morphism f . A curve
E of genus one over a field K tsuch that E(K) 6= ∅ has a structure of an
elliptic curve, a complete one-dimensional algebraic group. The structure
of an algebraic group is determined by a choice of a rational point that
serves as the unit element. With respect to this group structure the set
of sections aquires a structure of a group, the Mordell-Weil group of the
2.2. MODULAR SURFACES 35
• • Ã1 = I2 , III
• •
• •
.. ..
. . Ãn = In+1
• •
• •
1 2 2 2 2 1
• • • ... • • • ∗
D̃n = In−4
•1 •1
1 2 3 2 1
• • • • • Ẽ6 = IV ∗
•2
•1
1 2 3 4 3 2 1
• • • • • • • Ẽ7 = III ∗
•2
2 4 6 5 4 3 2 1
• • • • • • • • Ẽ8 = II ∗
•3
Figure 2.2: Reducible fibers of genus one fibration
elliptic surface. If the surface admits at least one singular fiber, the group
is a finitely generated abelian group.
Let us return to modular elliptic surfaces. It follows from the definition
of a universal family for a fine moduli space that that the Mordell-Weil group
of such a surface contains a subgroup isomorphic to (Z/N Z)2 . In fact, one
can show that there is nothing else. A fiber over a point x ∈ X(N )o is an
elliptic curve with level N whose isomorphism class is the point x. The level
structure on all such curves is defined by N 2 disjoint sections which form
a group isomorphic to (Z/N )2 . If N > 2, a fiber over a cusp is a singular
fiber of Kodaira’s type IN . The genus of a section C is equal to the genus
of X(N ). A section C0 of S(N ) → X(N ) is a curve isomorphic to the base
X(N ). Its genus is equal to
cN (N − 6)
g(X(N )) = 1 + .
12
Th self-intersection of a section C is equal to
C 2 = −cN N/12.
The geometric genus pg := h0 (OS(N ) ) of the surface S(N ) is equal
cN (N − 3)
pg (SN ) = .
6
Although for N = 2 a fine moduli space does not exist, one can define a
close approximation to it, an elliptic surface S(2)∗ → P1 with Mordell-Weil
isomorphic to (Z/2Z)2 . The surface S(2)∗ is obtained from the del Pezzo
surface D5 of degree 5 by the following construction. Let U be the union of
the 10 lines on D5 . Its divisor classe in the basis e0 , e1 , e2 , e3 , e4 of Pic(S)
defined by the class of e0 = π ∗ (OP2 (1)) and the classes of the exceptional
curves R1 , . . . R4 is equal to
4
X X
ei + (e0 − ei − ej ) = 6e0 − 2(e1 + e2 + e3 + e4 ). (2.2.1)
i=1 1≤i<j≤4
The double cover of S branched along the divisor U is defined locally by

zV2 − φV = 0, where φV = 0 are local equations of U in an affine open set
V . It can be glued to a global cover if the divisor class of U is divisible by
2 in the divisor class group Pic(D5 ). As formula (??) shows, this is satisfied
in our case. Let X → D5 be such a cover. If x ∈ U is a nonsingular point
of U, the local equation of U can be chosen to be u = 0, where u is a local
parameter. Thus over this point the equation of X is given by z 2 − u = 0,
it is a nonsingular open subset of X. On the other hand, if x is a singular
point of U, i.e. equal to one of the intersection points of two components of
U, the local equation can be chosen in the form uv = 0, thus z 2 − uv = 0
defined a surface that is singular of the point x. The singularity is as simple
as possible. It is called an ordinary double point, an ordinary node, or a
rational double point of type A1 . The elliptic surface S(2)∗ is obtained as a
minimal resolution σ : S(2)∗ → X of singularities of X. Its elliptic fibration
is the pre-image of one of five pencils of conics on D5 . They are the pre-
images on D5 of the pencils of of lines through one of the points p1 , . . . , p4 or
the pencil of conics through all of these points. The pre-images of singular
points of X on S(2)∗ are smooth rational curves Ci with self-intersection
−2 (a (−2)-curve, for short). A fiber of S → P1 over cusps 0, 1, ∞ consists
of two (−1)-curves intersecting at one point. It is also intersected by four
curves R1 , R2 , R3 , R4 . So, we see that X has 15 singular points, five over
2.3. NORMAL ELLIPTIC CURVES 37
each fiber. The pre-image of the fiber of X → S → P1 over a cusp is a

fiber of Kodaira’s type I3∗ . The elliptic surface S(2)∗ is a K3 surface. i.e.
an algebraic surface X satisfying KX = 0 and H 1 (X, OX ) = {0}. It is
characterized (among elliptic surfaces) by the property that B ∼ = P1 and
X
e(f −1 (b)) = 24, (2.2.2)
b∈B
where e() denotes the topological Euler-Poincaré characteristic (computed

in étale topology if k 6= C).
The choice of a zero section defines a group law on each nonsingular fiber.
The involutions y 7→ −y on each fiber are glued together to define a global
involution T : S(N ) → S(N ), the quotient Kum(N ) = S(N )/(T )by this
involution is called the universal Kummer family. It comes with a projection
map Kum(N ) → X(N ) whose fibers over points in X(N )o are isomorphic
to P1 . For N > 2, the fibers over cusps are chains of k = [N + 2/2] smooth
rational curves R1 + R2 + · · · + Rk with R12 = Rk2 = −1, Rj2 = −2, j 6= i, k
and Ri · Ri+1 = 1, i = 1, . . . , k − 1, all other intersections are zeros. Starting
from R1 we blow down the curves in each fiber until we get a minimal ruled
surface.
2.3 Normal elliptic curves

Again we assume here that k is algebraically closed. Let Q1 , Q2 be two
quadrics in P3 such that C = Q1 ∩ Q2 is a smooth curve. Projecting from
a point x0 ∈ C to P2 , we find that the image of C is a plane cubic. It is a
curve of genus 1, an elliptic curve. The same fact follows from the adjunction
formula. If we fix a smooth quadric Q in the pencil spanned by Q1 , Q2 (we
will see later that this is always possible if the intersection is smooth), then
C is a divisor on Q given by a section of OQ (2). The canonical sheaf of
P3 is OP3 (−4), and the adjunction formula for Q gives ωQ = OQ (−2), then
the canonical sheaf of C is OC . This property is the definition of an elliptic
curve.
Let E be an elliptic curve and p0 be the point on E which we choose to
be the zero point in the group law. By Riemann-Roch,
h0 (np0 ) := dim H 0 (E, OE (np0 )) = n.
Let (s0 , . . . , sn−1 ) be a basis of the space H 0 (E, OE (np0 )). Its elements have
a pole of order n at p0 and vanish at some set (si )0 of n points in E. Assume
n ≥ 3. The map
φ : E → Pn−1 , x 7→ [s0 (x), . . . , sn−1 (x)]
is a projective embedding of E in Pn−1 .

Note that the linear system |np0 | of effective divisors linearly equivalent
to np contains n2 divisors n(p0 ⊕ e), where ⊕ denotes the addition in the
group law and e ∈ E[n] is a n-torsion point. They must be cut out by
hyperplanes in Pn−1 . They are called osculating hyperplanes. They intersect
the image of E in Pn−1 at one point with multiplicity n.
Assume that (n, char(k)) = 1. An elliptic curve E has n2 points x such
that nx = 0 in the group law. The group E[n] of such points is isomorphic
to (Z/nZ)2 . If k = C, then E = C/Z + Zτ, where τ = a + bi, b > 0. The
n-torsion points are the cosets of the point n1 (k + mτ ), 0 ≤ k, m ≤ n − 1.
Let Gn = (Z/nZ)2 . Consider its projective representation in a vector
space V with a basis (e0 , . . . , en−1 ) that is given on generators σ = (1, 0)
and τ = (0, 1) by the formual
σ : ei → ei−1
τ : ei 7→ in ei ,
where i ∈ Z/nZ and n is a generator of the group µn (k) = {a ∈ k : an =

1}. It is an irreducible projective representation that originates from the
Schrödinger representation of the Heisenberg group Hn that fits in the exact
sequence
1 → µn → Hn → (Z/nZ)2 → 1.
Here [σ, τ ] = n id, so group Hn is not commutative but rather nilpotent.
Proposition 2.1. There exists a unique basis in H 0 (E, OE (np0 )) such that
the embedding of E in Pn−1 defined by this basis is invariant with respect to
the Schrödinger representation.
Proof. Since under the addition map τα : p 7→ p ⊕ α, where α ∈ E[n], the
divisor class np does not change, we see that invertible sheaf L = OE (np)
satisfies τα∗ (L) ∼
= L. Let φα be an isomorphism τα∗ (L) → L. It is defined
uniquely up to an automorphism of L which is given by a scalar automor-
phism c : OE → OE . It is immediately checked that the set of automor-
phisms (φα ) satisfies
φα+β = cα,β φ∗β (τα ) ◦ φα ,
where cg,g0 is a nonzero constant. The set (cα,β )α,β∈G is a 2-cocycle whose
cohomology class belongs to H 2 (G, k∗ ). Computing the cohomology group,
we find that this cocycle can be chosen to be equal to the cocycle defined by
(α, β) 7→ [α, β]. By a general yoga of group cohomology, this implies that
the projective representation ρ : G → PGL(H 0 (E, L)) defined by ρ(α)(s) =
φα (τα∗ (s)) lifts to a linear representation of Hn in H 0 (E, L). It follows from
the definitions of the sheaves τα∗ (L), that it is compatible with the action of
its quotient G on E. The latter means that, for all s ∈ H 0 (E, L) and x ∈ E,
h∗ (s)(x) = s(h̄−1 (x)),
where h̄ is the image of h in G.

Now, we see that there exits a central extension Ẽ[n] of E[n] isomorphic
to Hn that admits a linear representation in H 0 (E, L). By means of the
dual representation, it acts in the projective space P(H 0 (E, L)) ∼ = Pn−1 in
which E is embedded by means of the linear system |np0 |. The action factors
through the group E[n] and defines a projective representation of E[n] in
Pn−1 .
One can show that the Schrödinger V representation of Hn is a unique
(up to isomorphism) irreducible linear representation on which the subgroup
µn acts in a natural way by scalar multiplication. It also can be character-
ized by the property that, for any cyclic subgroup of order n of Hn that
projects isomorphically to G, the linear subspace V K of invariant elements
is one-dimensional. Our representation of E[n] on H 0 (E, OE (np0 )) satisfied
these properties. The last property is shown as follows. We choose a cyclic
subgroup K of E[n] that can be isomorphically lifted to a subgroup of Ẽ[n].
Under an isomorphism Ẽ[n] → Hn , it can be taken to be the subgroup hσi
or hτ i. Let f : E → E/K be the quotient map. The curve E 0 = E/K is an
elliptic curve isogenous to E. Since L admits a linearization with respect to
K, we see that L = f ∗ M, for some invertible sheaf M. Since deg L = n,
we have deg M = 1. By Riemann-Roch, dim H 0 (E 0 , M) = 1. Also, we have
H 0 (E 0 , M) = H 0 (E, L)K .
Thus we have proved that one can choose an isomorphisms φ : Hn →
Ẽ[n] that descends to a level n-structure Gn = (Z/nZ)⊕2 → E[n] such that
the linear action of Ẽ[n] on H 0 (E, OE (np0 )) is isomorphic to the Schrödinger
representation of Hn in V .
Remark 2.2. What we have glimpsed in during this proof is the general
theory of linearization of an action of an algebraic group G on an algebraic
variety X (not necessary faithful). It is a choice of a sheaf F such that there
exist isomorphisms φg : g ∗ (F) → F, g ∈ G satisfying φg0 ◦g = φ∗g0 (φg ) ◦ φg .
This allows to define a linear representation of G in the space of sections of

F. We refer for the details to [Mumford, Geometric Invariant Theory].
Let [−1] : E → E be the negation involution of an elliptic curve with
respect to its group law. It satisfies
[−1] ◦ ta = t−a ◦ [−1], (2.3.1)
where ta : E → E is the translation map x 7→ x ⊕ a. Let V be the vector

space H 0 (E, OE (np0 )) with a chosen basis (e0 , e1 , . . . , en−1 ) as above. Since
[−1](p0 ) = p0 , the sheaf OE (np0 ) admits a canonical linearization with
respect to [−1] (in fact, one can show that an invertible sheaf OC (D)) on
an algebraic curve C on which a finite group G acts admits a linearization
with respect to G if and only if g(D) = D for all g ∈ G). Let ι : V → V be
the linear action of [−1] on V . The identity (??) gives, for all g ∈ G.
ι ◦ g = g −1 ◦ ι.
It is easy to write the possible matrix of ι satisfying these identity. We get
ι : ti 7→ t−i .
The linear space V decomposes into two eigensubspaces V ± with respect to

ι with eigenvalues ±1. We have
V + = he0 , e1 + en−1 , e2 + en−2 , . . . , e[ n2 ] + e[ n+1 ] i,

2
V − = he1 − en−1 , e2 − en−2 , . . . , e[ n2 ] − e[ n+1 ] i,

2
where the last vector in V − should not be ignored if n is even. In particularly,

we find that
n n
dim V + = 1 + [ ], dim V − = n − 1 − [ ].
2 2
If n = 2k + 1 is odd, we get dim V ± = 12 (n ± 1). If n = 2k is even, we get
dim V + = 1 + k, dim V + = k − 1. Let Vg− = g(V − ), g ∈ G. We have n2 such
subspaces, and hence n2 corresponding linear projective subspaces in P(V )
of dimension equal to dim V − − 1. Let C be a coset of some cyclic subgroup
of E[n]. The image of C in P(V ) consists of n points that span a hyperplane
HC .
Assume that n is an odd number. There are n + 1 cyclic subgroups and
n(n+1) cosets. Thus we have a configuration of n(n+1) hyperplanes HC and
p2 subspaces P(Vg− ) of dimension 12 (n − 3). Each HC contains n subspaces

P(Vg− ) and each P(Vg− ) is contained in n+1 hyperplanes HC . This realizes an
abstract configuration (n(n + 1)n , n2n+1 ). An abstract configuration (ac , bd )
is a relation on two sets A, B of cardinalities a, b such that each element of A
is related to c elements of B, and each element of B is related to d elements
in A. The subspaces P(V − ) intersects φ(E) at the unique fixed point, the
origin (here we use that n is odd). The other subspaces P(Vg− ) intersects
φ(E) at other n-torsion points.
When k = C, one constructs the basis in H 0 (E, OE (np)) explicitly by
using the theory of Riemann theta functions or Weierstrass σ-functions on
C. Let
z z z2
(1 − )e r + 2r2 ,
Y
σ(z) := z
r
r∈Λ\{0}
where Λ = Z + Zτ . It satisfies
1
σ(z + 1) = −eη1 (z+ 2 ) σ(z),
τ
σ(z + τ ) = −eη2 (z+ 2 ) σ(z).
for some η1 , η2 ∈ C. Let

p + qτ
σpq (z) := σ(z − ), p, q ∈ Z,
n
n−1 η2 ω1 η1
ω1 := −e− 2 n , ω2 := e− 2n .
Then one defines the sections sm , m = 0, . . . , n − 1 by
2
sm = ω1m ω2m emη1 z σm,0 (z) · · · σm,n−1 (z).
Example 2.3. Assume n = 3. Then the ring of invariant polynomial

P (t0 , t1 , t2 ) with respect to the Schrödinger representation is generated by
t30 , t31 , t32 , t0 t1 t2 and the equation of φ(E) in P2 (after scaling the coordinates)
is a Hesse equation
t30 + t31 + t32 + λt0 t1 t2 = 0,
where 8λ3 + 1 6= 0 if the cubic is nonsingular. The osculating hyperplanes
in this case are nine flex tangent lines that are tangent to the cubic curve at
one point. The corresponding points are inflection points of the cubic. Their
coordinates do not depend on the parameter and they lie on the coordinate
lines ti = 0. We may assume that the points on the line t0 = 0 form
a subgroup H of E[3]. Then points on the other two lines form two non-
trivial cosets with respect to this subgroup. There are 4 proper subgroups of
G, and there are 12 cosets altogether. Each contain three collinear inflection
points. The corresponding lines are given by equations
x = 0, y = 0, z = 0, Hi,j := t0 + i3 t1 + j3 t2 = 0, 0 ≤ i, j ≤ 2.
The two sets of 9 points and 12 lines form the famous abstract Hesse con-
figuration (123 , 94 ). There are 4 singular members in the Hesse pencil. One
is given by equation t0 t1 t2 = 0 corresponding to the parameter λ = ∞.
Other fibers are also triangles of lines, they correspond to the parameters
λ satisfying 8λ3 + 1 = 0. The blow-up of 9 base points of the pencil is the
modular elliptic surface S(3). The elliptic fibration is given by the resolving
the indeterminacy points of the rational map
P2 99K P1 , [t0 , t1 , t2 ] 7→ [t30 + t31 + t32 , t0 t1 t2 ].
Its singular members are of Kodaira’s type I3
Example 2.4. Assume n = 4. Let φ(E) = C ⊂ P4 = |V ∨ |, where V =

H 0 (E, OE (4p)), be invariant quartic curve with respect to the Schrödinger
representation. Let Q be a quadric in P4 , it either contains C or cuts out
a divisor of degree 8 on C equal to the zero scheme of a section of OC (2).
By Riemann-Roch, h0 (OC (2)) = 8. Since there are 10 linear independent
quadratic forms in 4 variables, we see that the restriction homomorphism
r : H 0 (C, OP3 (2)) → H 0 (C, OC (2))
has two-dimensional kernel. Thus there exists two linear independent quadrics
Q1 = V (q1 ), Q2 = V (q2 ) that contain C. Since the degree of Q1 ∩Q2 is equal
to 4, we obtain that C = Q1 ∩ Q2 .
Since C is invariant with respect to the Schrödinger representation, the
pencil hq1 , q2 i must be invariant two with respect to the representation of
the Heisenberg group in the symmetric square of V = H 0 (C, OC (2)) ∼ = k4 .
It is easy to see that
S 2 V = ⊕5i=1 Vi ,
where Vi are irreducible sub-representations given explicitly by
V1 = ht20 + t22 , t21 + t23 i, V2 = ht20 − t22 , t21 − t23 i, V3 = ht0 t3 , t1 t2 i,
V4 = ht0 t1 + t2 t3 , t0 t2 + t1 t3 i, V5 = ht0 t1 − t2 t3 , t0 t2 − t1 t3 i.
Note that V1 and V3 are isomorphic representations, the generators σ and τ

act similarly on their bases. So, our quartic curve C must be the intersection
of two quadrics either spanning V4 or V5 or to be contained in V1 ⊕ V3 . The
two quadrics can not spend V4 or V5 since we find, looking at the matrix
of partials, that the intersection point [1, 0, 1, 0] will be a singular point. It
cannot be equal neither to V2 nor to V1 or V3 . It is easy to see that the only
possibility is that
C(λ) : t20 + t22 + λt1 t3 = t21 + t23 + λt0 t2 = 0.
The parameter λ here satisfies
λ(λ4 = 1) 6= 0, λ 6= ∞.
Thus there are six singular members in the family. Each is isomorphic to
the union of four lines forming a quadrangle. In the case n = 4, we have
n + 1 obvious cyclic subgroups generated by (1, 0), (1, 1), (1, 2), (1, 3), (0, 1)
of G4 = (Z/4Z)⊕2 as well as a new subgroup generated by (2, 1). So, the
configuration of cosets HC and subspaces P(Vg− ) realizes an abstract config-
uration (244 , 166 ). Note that, since n is even in this case, the point P(V − )
is not on the embedded curve φ(E). The fixed points of the involution [−1]
are four 2-torsion points, and they all lie in the hyperplane P(V + ) which
intersects the quartic curve at four points.
Let S be the union of all curves C(λ), λ ∈ P1 . Fix a line ` in P3 . The
quadrics Q1 (λ) intersect ` at two points, and then λ moves in P1 and form
a linear series of degree 2. Another degree 2 series of degree 2 is formed by
intersecting Q2 (λ) with `. The graphs of the corresponding maps of degree
2 ` → P1 are curves of bi-degree (2, 1) on P1 × P1 . They intersect at 4
points. This implies that ` intersects four curves in the family, and hence S
is a quartic surface in P3 . It is a modular elliptic surface S(4) of level 4. It
contains 6 singular fibers of Kodaira’s type I4 , the modular cuve X(4) is of
genus 0, hence, applying formula (??) we obtain that S(4) is a K3 surface.
Example 2.5. Finally, assume that n = 5. Here we skip the details referring
to [Hulek, Asterisques]. The curves C are given by intersection of 5 quadrics
in P4 .
Q0 = t20 + λt2 t3 − λ−1 t1 t4 = 0,

Q1 = t21 + λt3 t4 − λ−1 t0 t2 = 0,
Q2 = t22 + λt0 t4 − λ−1 t1 t3 = 0,
Q3 = t23 + λt0 t1 − λ−1 t2 t3 = 0,
Q4 = t24 + λt1 t2 − λ−1 t0 t3 = 0,
These are normal quintic elliptic curves. One can show that C(λ) is equal to
the intersection of the Grassmann variety G(2, 5) ⊂ P9 with a linear subspace
of dimension 4. The linear subspaces of dimension 4 in P9 are parameterized
by the Grassmann variety G(5, 10) of dimension 25. The group of projective
automorphisms of P5 is of dimension 24. Thus the number of projective
moduli of elliptic quintics is equal to 1, as expected. There are 12 singular
curves among C(λ). Each consists of a pentagon of lines. The union of the
curves C(λ) is an elliptic modular surface S(5). The parameter a belongs to
the modular curve X(5) of genus 0. The surface S(5) has pg := h0 (ωS(5) ) =
4.
One can prove that, for any n ≥ 4, a normal elliptic curve of degree n is
given by a linear system of quadratic equations.
Lecture 3
Pencils of quadrics
3.1 Discriminant hypersurface

Let |S 2 E ∨ | be the projective space of quadrics in |E| ∼ = Pn . The reduced
2 ∨
subvariety Dn of |S E | that consists of singular quadrics is called the dis-
criminant hypersurface. Choose a basis e1 , . . . , en+1 in E with the dual
basis (t1 , . . . , tn+1 ) in E ∨ . The basis in S 2 E ∨ consists of monomials ti tj and
the coordinates in S 2 E ∨ are generic coefficients Aij of quadratic forms. If
char(k) 6= 2, the equation of the discriminant hypersurface is
det(Bij ) = 0,
where Bii = 2Aii and Bij = Bji = Aij . It is a hypersurface of degree n + 1.
If n is even, and char(k) = 2, then the equation of Dn is
X
Aij Pf(A)i Pf(Aj ) = 0,
1≤i≤j≤n+1
where Pf(A)i is the Pfaffian of the matrix obtained from the matrix (Bij )
from above by deleting the ith row and the ith column. Again, this is a
hypersurface of degree n + 1. If n is odd and char(k) = 2, we take for the
equation of Dn the Pfaffian of the matrix B = (Bij ). This is a hypersurface
of degree 12 (n + 1).
Let us look at the singularities of Dn . We assume here that char(k) 6= 2.
Let D̃n be the subscheme of |S 2 E ∨ | × |E| given by n + 1 bi-homogeneous
equations
B · t = 0,
where t is the column-matrix with entries (t1 , . . . , tn+1 ). It is clear that its
k-points are pairs (Q, x), where Q is a quadric and x is its singular point.
45
46 LECTURE 3. PENCILS OF QUADRICS
There are two projections
D̃n .
p1 p2
~
Dn |E|
The fiber of the first projection over a quadric Q is its singular locus, a linear
subspace of |E|. The fiber of the second projection over a point x ∈ |E| is a
linear subspace of |S 2 E ∨ | of quadrics that contain x in its singular locus. It
is a linear projective subspace of dimension equal to dim |S 2 E ∨ | − n − 1 =
n+1

2 − n − 1. It follows that the scheme D̃n is smooth of dimension equal
to dim |S 2 E ∨ | − 1 = 21 (n2 + 3n − 2). Thus the projection p1 is a resolution of
singularities of the discriminant variety, i.e. a proper morphism of a smooth
scheme which is an isomorphism over an open subset of smooth points.
The following proposition follows from the theory of determinantal va-
rieties, we omit the proof.
Proposition 3.1. Let SMm (k) be the affine variety of symmetric matrices
of size m and corank ≥ k. Then
• SMm (k) is an irreducible Cohen-Macaulay subvariety of codimension

1
2 k(k + 1);
• Sing(SMm (k)) = SMm (k + 1);

m+i

Q k−i
• deg SMm (k) = 0≤i≤k−1 2i+1 .
i
This gives the stratification of the singular locus Sing(Dn ) of the dis-
criminant hypersurface:
Sing(Dn ) = Sing(Dn )0 ⊃ Sing(Dn )1 ⊃ . . . ⊃ Sing(Dn )n−1 ⊃ ∅,
where Sing(Dn )t is the closed subvariety parameterizing quadrics Q with

singular locus of dimension t. Here each Sing(Dn )t is the singular locus of
Sing(Dn )t−1 . We also have
1 1
codim(Sing(Dn )t , Dn ) = (t + 1)(t + 2) − 1 = t(t + 3).
2 2
We can also describe the tangent space of Dn at its smooth point Q0 . Let
Sing(Q) = {x0 }. Since p1 is an isomorphism over Q0 , the tangent space is
3.1. DISCRIMINANT HYPERSURFACE 47
isomorphic to the tangent space of D̃n at the point (Q0 , x0 ). The description
of the second projection shows that it is isomorphic to the tangent space of
the fiber of p2 over the point x0 (it is certainly the subspace of the tangent
space and its dimensions agree, so it must be the whole space). Thus we
obtain
Proposition 3.2. Let Q0 be a smooth point of Dn . Then the tangent space

of Dn at the point Q0 is naturally isomorphic to the space of quadrics that
contain the singular point of Q0 .
The projective space |S 2 E ∨ | is the complete linear system of quadrics

|O|E| (2)| in the projective space |E| = P(E ∨ ). Let |L| ⊂ |S 2 E ∨ | is a linear
system of quadrics in |E| of dimension r. If r = 1, 2, 3, > 3, we say that |L|
is a pencil, a net, a web, a hyperweb.
Let D(|L|) be the intersection of |L| with the discriminant hypersurface
Dn . We say that L is a regular if |L| intersects Dn transversally. This means
that, for all t ≥ 0, the intersection |L| ∩ (Sing(Dn )t \ Sing(Dn )t+1 ) is smooth
of codimension in |L| equal to the codimension of Sing(Dn )t in |S 2 E ∨ |. This
is true when |L| is chosen to be generic, i.e. belongs to some open subset in
the Grassmannian G(r + 1, |S 2 E ∨ |).
Example 3.3. Let n = 2. Assume char(k) 6= 2. The discriminant hyper-

surface D2 is a cubic hypersurface in P5 given by equation
 
2A11 A12 A13
det  A12 2A22 A23  = 0.
A13 A23 2A33
Its singular locus is the Veronese surface of degree 4, the image of the map
P2 → P5 given by
(x, y, z) 7→ [A11 , A22 , A33 , A12 , A13 , A23 ] = [x2 , y 2 , 2z 2 , 2xy, 2xz, 2yz].
Assume p = 2. Then the discriminant hypersurface is again a cubic

hypersurface. Its equation is
A11 A223 + A22 A213 + A33 A212 + A12 A23 A13 = 0.
Its singular locus is given by equations A12 = A13 = A23 = 0. It is a linear

subspace of dimension 2.
3.2 Pencils of quadrics

Assume r = 1, i.e. |L| is a pencil of quadrics. Let |L| be generated by
quadrics Q1 , Q2 and X = Q1 ∩ Q2 be the base locus of |L|. The line |L|
intersects the hypersurface Dn at deg Dn = n + 1 points counting with mul-
tiplicities.
Theorem 3.4. Assume char(k) 6= 2. The following properties are equiva-
lent:
(i) The variety X is nonsingular;
(ii) Dn (|L|) consists of n + 1 distinct points.
(iii) X is projectively isomorphic over k̄ to the variety defined by the equa-

tions
n+1
X n+1
X
t2i = ai t2i , (3.2.1)
i=1 i=1
where ai 6= aj , i 6= j.
Proof. (i) ⇒ (ii). Suppose |L| consists of less than n + 1 points. This means
that |L| either intersects Dn at its singular point, or belongs to the tangent
space of Dn at its nonsingular point. In the first case |L| contains a quadric
Q with dim Sing(Q) ≥ 1. This implies that any other quadric Q0 in the
pencil intersects Sing(Q) at some point x. Let Tx (X) denote the tangent
space of a projective subvariety X ⊂ Pn at its point x. It is a projective
subspace of Pn of dimension equal to dimx (X) if x is a nonsingular point of
X. We have
Tx (X) = Tx (Q) ∩ Tx (Q0 ) = Tx (Q0 ), (3.2.2)
hence dim Tx (X) is larger than the dimension of X (equal to n − 2). Thus
x is a singular point of X. This contradiction proves the implication.
(ii) ⇒ (i) Suppose X is singular at some point x. Then, it follows from
(??) that any two smooth quadrics in |L| are tangent at some point x (i.e.
their tangent spaces at this point coincide). If we write Q = V (q), Q0 =
V (q 0 ), x = [v], then this means that the linear forms bq (v) and bq0 (v) are
proportional (recall, that we view a bilinear form as a linear map E → E ∨ ).
Thus, a linear combination of q, q 0 defines a quadric Q that is singular at
the point x. So, we see that all quadrics in |L| contain a singular point of
Q. Applying Proposition ??, we find that |L| is contained in the tangent
space of Dn at the point Q, and hence it intersects Dn at Q with multiplicity
larger than 1. This contradiction proves the implication.
3.3. SEGRE’S SYMBOL 49
(ii) ⇒ (iii) We know that a nonsingular quadric can be reduced to the

form given by one of the equations (??). Since char(k) 6= 2, we can reduce
any of these two equations to the sum of squares n+1 2 . Since |L| contains
P
t
i=1 i
a nonsingular quadric Q1 , we can reduce its equation to this form. Let A
be the matrix of the bilinear form bq for some other nonsingular quadric
Q2 = V (q) different from Q1 . Then D(|L|) consists of points [1, λ] ∈ P1
such that det(A − λIn+1 ) = 0. Since it consists of n + 1 different points, we
see that the matrix A has n + 1 distinct eigenvalues. Since A is symmetric,
there exists a matrix C such that C t · A · C is a diagonal matrix Λ with
distinct entries (belonging to k̄) at the diagonal. To do this, we first find
S such that S −1 · A · S = Λ, then use that A is symmetric to obtain that
(t S · S) · Λ · (t S · S)−1 = Λ, hence t S · S is diagonal, and then scale the
coordinates to obtain that (t S · S) = In+1 . Thus we can find a basis such
that the quadrics Q1 and Q2 are given by the equation in the assertion (iii).
(iii) ⇒ (i) Obviously the quadrics V (ai q1 − q2 ) are singular with isolated
singular point, and the number of them is equal to n + 1.
3.3 Segre’s symbol

The classification of pencils of quadrics with singular base locus is given by
the Segre symbol
(1) (s ) (1) (s )
[(e1 . . . e1 1 )(e2 . . . e2 2 ) . . . (e(1) (sr )
r . . . er )].
For any e ≥ 1 and α ∈ k consider the following quadratic forms:

e
X e−1
X
p(α, e) = α ti te+1−i + ti+1 te+1−i ,
i=1 i=1
e
X
q(e) = ti te+1−i .
i=1
Theorem 3.5. Assume char(k) 6= 2 and k is algebraically closed. Assume

that one of the quadrics in the pencil is nonsingular. Any pair of non-
proportional quadratic forms q1 , q2 can be written in some coordinates in the
form
si
r X
(j)
X
q1 = p(αi , ei ),
i=1 j=1
r X si
(j)
X
q2 = q(ei ),
i=1 j=1
Proof. A proof can be found, for example, in [Hodge-Pedoe, vol. 2] or

[Gantmacher]. Let us give it, for completeness sake.
Let R = k[t] (or any principal ideal ring) and M be matrix with entries in
R. Recall that the definition of elementary divisors of M (see, for example,
[Gelfand, Linear Algebra]). Let pk be monic polynomials equal to the great-
est common divisors of minors of M of size k. We have P1 |P2 | · · · Pn where
n is the smallest of the numbers of rows or columns of M . Each Pk is the
product of some powers of irreducible polynomials. These powers collected
from each Pk are called the elementary divisors of M . Let T : E → E be an
endomorphism of a linear space of dimension n over k. Define a structure
of a R-module on E by setting t · v = T (v), v ∈ E. A fundamental theorem
about modules over a principal ring (that can be found in many books on
abstract algebra) gives that E is isomorphic to the direct sum of modules
e e
of the form R/(pi j ), where pi j are elementary divisors of the characteristic
matrix t − A, where A is the matrix of T in some basis of E. Note that the
product of elementary divisors of a square matrix is equal to the determinant
of the characteristic matrix.
Let E[t] = k[t] ⊗k E be the R-module obtained from E by extension of
scalars. Let We have an exact sequence of R-modules
ψ φ
0 → E[t] → E[t] → E → 0,
where
ψ(f (t) ⊗ v) = tf (t) ⊗ v − f (t) ⊗ T (v), ψ(f (t) ⊗ v) = f (T )(v).
It is immediately checked that the sequence is exact (or, see [Bourbaki,

Algebra, Chapter 3]). So, one can interpret E as the cokernel of the map of
free modules over R defined by the endomorphism t − T . Choosing a basis
in E, we identify T with a matrix A, and the endomorphism is given by the
characteristic matrix t−A. The decomposition of E into the sum of primary
k k
modules of the form k[t]/(pi j ), where pi j are the elementary divisors of the
characteristic matrix t − A.
Let us apply this to our situation. We choose a basis (q1 , q2 ) of L and
a basis in E such that bq1 , bq2 are defined by symmetric matrices A, B such
that A is invertible and consider the endomorphism of E given by the matrix
A−1 · B. The characteristic matrix t − A−1 · B obtained from the matrix
At − B by multiplication by a scalar matrix. It is easy to see the elementary
divisors of the matrix At − B doe not depend on a choice of a basis in L nor
do they depend on a choice of a basis in E. Thus they are invariants of the
pencil.
3.3. SEGRE’S SYMBOL 51
We shall show that two pencils are projectively equivalent if their elemen-
tary divisors are the same. The assertion of the theorem will follow, since
the canonical forms represent all possible collections of elementary divisors.
Suppose two pencils defined by two pairs of symmetric matrices (A, B)
and (A0 , B 0 ) as above have the same elementary divisors. Then they define
isomorphic structures of k[t]-module on E. Let S be the matrix of this
transformation. It satisfies A−1 ·B = S −1 ·A0−1 ·B 0 ·S. Let P = A·S −1 ·A0−1 .
Then, we have
B = P · B 0 · S, A = P · A0 · S.
It is not what we want, we need a matrix Q such that t Q · A · Q = A0 and

t Q · B · Q = B 0 . It will define the needed projective transformation. Let us
find such a matrix.

Since A, B, A0 , B 0 are symmetric, we get
t
S · B 0 · t P = B, t
S Ȧ0 · t P = A.
This implies that

(t S · P −1 ) · A · (S −1 · t P ) = A
and similar equality for B. Let M = t S · P −1 . We have
M · A = A · t M, M · B = B · t M.
Obviously, we have f (M ) · A = A · t f (M ), f (M ) · B = B · t f (M ) for any

polynomial f (t). Choose f (t) such that f (M )2 = M . It is always possible
2
since the matrices In+1 , . . . , M (n+1) are linearly dependent. Let N = f (M ).
Then, we have N · A = A · t N, N · B = B · t N , hence
A0 = P −1 · A · S −1 = t S −1 · M · A · S −1
= t S −1 · N 2 · A · S −1 = (t S −1 · N ) · A · (t N · S −1 ).
Now the matrix S 0 = t S −1 · N is the matrix that does the job. This proves
the assertion.
Note that if D(|L|) has n + 1 distinct roots, the elementary divisors are
all linear polynomials, and the canonical form coincides with the form given
in Theorem ??. This gives another proof of the implication (i)⇒ (iii).
3.4 The case when the characteristic is equal to 2

Here we only state the results. First assume that n is odd. In this case the
discriminant of the pencil is given by the pfaffian of the matrix of even size
n + 1 with entries homogeneous linear forms in two variables. It is expected
to have 12 (n + 1) zeroes.
The following result belongs to Usha Bhosle (Crelle Journal, vol. 407).
Theorem 3.6. Assume that the discriminant of the pencil is a reduced
polynomial of degree k = 12 (n + 1). Then, on some coordinates, the pencil is
generated by quadrics V (q1 ) and V (q2 ), where
k
X
q1 = t2i−1 t2i ,
i=1
Xk
q2 = ai t2i−1 t2i + ci t22i−1 + di t2i ,
i=1
where a1 , . . . , ak are the roots of the discriminant.

Qk The base locus X =
V (q1 ) ∩ V (q2 ) is nonsingular if and only if i=1 ci di 6= 0.
Suppose n is even. Then a recent result of myself and Alex Duncan is
the following:
Theorem 3.7. Let (q1 , q2 ) be a pair of quadratic forms on a vector space
E of dimension n = 2m + 1 ≥ 3 over a field k of characteristic 2. Suppose
that the intersection of quadrics V (q1 ) ∩ V (q2 ) is smooth. Then there exists
a basis (x0 , . . . , xm , y0 , . . . , ym−1 ) in E ∨ such that
m
X m−1
X m−1
X
q1 = a2i x2i + xi+1 yi + r2i+1 yi2 ,
i=0 i=0 i=0
(3.4.1)
m
X m−1
X m−1
X
q2 = a2i+1 x2i + xi yi + r2i yi2 ,
i=0 i=0 i=0
where the coefficients a0 , . . . , an are equal to those of the half-discriminant

polynomial, and r0 , . . . , rn−2 are in k. If k is algebraically closed, one can
choose ri ’s to be zero.
3.5 Intersection of two quadrics in P2n

Let X = Q1 ∩ Q2 be a smooth intersection of two quadrics in P2k . It is
a subvariety of P2k of dimension 2k − 2 and degree 4. We have already
3.5. INTERSECTION OF TWO QUADRICS IN P2N 53
considered the case k = 1, where X consisted of four distinct points in the

plane. The next case is k = 2, where X is an algebraic surface in P4 . By
the adjunction formula, ωX ∼ = OX (−1), it is a del Pezzo surface of degree 4.
As we explained in the previous Lecture, X is isomorphic to the blow-up of
5 points p1 , . . . , p5 in the plane. The anti-canonical class is the divisor class
3e0 − e1 − · · · − e5 . Since it is ample, it has positive intersection with any
curve on X. In particular, no three points pi lie on a line since otherwise the
proper inverse image of this line belongs to the divisor class e0 − ei − ej − ek ,
and hence does not intersect −KX . We also explained how to see lines on
X. Since X = X ac , each line comes from a (−1)-curve on X, and the latter
are obtained as proper inverse transforms of a line hpi , pj i, or of the conic
through p1 , . . . , p5 , or of the exceptional curve over one of the points pi .
Altogether we have 16 lines. They intersect each other according to the
following graph.
•
• • •
• •
• • • •
• •
• •
•
•
Figure 3.1: Lines on a del Pezzo quartic surface
Note that a birational morphism σ : X → P2 is defined by a choice of

five skew lines in X. Inspecting the graph, we find that there are 5! of such
choices. Fix a line ` on X. Then there will be exactly five disjoint lines
`1 , . . . , `5 intersecting ` (you can easily verify it by taking ` to be the proper
inverse image of the conic through the five points). Let π : X → P2 be the
blow-down of these lines. The image of ` will be a conic K. Fix a point
x0 ∈ ` not belonging to the set S of the intersection points of ` with other
five lines. For any other point x ∈ ` \ S, consider the line ` = hπ(x0 ), π(x)i
in the plane. The union of this line and the conic is a cubic curve through
the five points pi , the images of the lines ì . It corresponds to a hyperplane
section of the anti-canonical model X. Since it is singular at the point x0 ,
it contains the tangent space Tx0 X. There will be a unique nonsingular
quadric Q that contains this hyperplane as its tangent space at x0 . Also, if
x ∈ S and equal to the intersection point ` ∩ ì , then the restriction of any
quadrics to the plane spanned by ` and ì contains the same conic ` ∪ ì ,

hence there will be a quadric Qi that contains the whole plane. Since a
nonsingular quadric does not contain planes, Qi is one of the five singular
quadrics in the pencil. This shows that we can identify ` with |L|. Under
this identification, the set {` ∩ `1 , . . . , ` ∩ `5 } is identified with the D(|L|).
Thus, via the Veronese map whose image is the conic through p1 , . . . , p5 , the
image of D(|L|) is equal to the set of points {p1 , . . . , p5 }.
Let π : X → P2 be the blowing down of 5 disjoint lines `1 , . . . , `5 and
let e0 , e1 , . . . , e5 be the corresponding basis in Pic(X). Since KX = 3e0 −
(e1 + · · · + e5 ), we see that the orthogonal complement of KX has a basis
(e0 − e1 − e2 − e3 , e1 − e2 , e2 − e3 , e3 − e4 , e4 − e5 ) (it is called a geometric
basis). Computing the intersection matrix, we find that it coincides with
the Cartan matrix of the root system of type D5 . Fixing an order on the
set of `1 , . . . , `5 , defines a basis in Pic(X). A del Pezzo surface with a fixed
ordered set of five skew lines is called a marked quartic del Pezzo surface.
We see from the previous discussion that a marking of a quartic del Pezzo
surface is equivalent to any of the following:
• a choice of an ordered set of five skew lines on X;
• a choice of an order on the set of singular quadrics in the pencil of

quadrics containing X;
• a choice of a geometric basis in Pic(X);
• a choice of an isomorphism between the subgroup KX ⊥ ⊂ Pic(X)
equipped with the intersection product quadratic form and the root
lattice of a simple Lie algebra of type D5 ;
• a choice of a bijection of the graph of 16 lines on X together with the

incidence relation defined by the intersection of lines and the graph
given in Figure ??.
Thus, the set of isomorphism classes of marked quartic del Pezzo surfaces
of degree 4 is equal to the set of PGL(3)-orbits of 5 ordered points in P2 nor
three of which are collinear. Via the Veronese map it is the same as the set
of PGL(2)-orbits of 5 ordered points in P1 . Since we can always fix the first
four points by a projective transformation, we obtain that the moduli space
of marked quartic del Pezzo surfaces can be identified with the open subset
of a del Pezzo surface of degree 5 with the complemet equato the set of 10
lines on it. It coincides with the moduli space M0,5 . Thus, we see that the
same space could serve as the moduli space for different moduli problems.
3.6. INTERSECTION OF TWO QUADRICS IN AN ODD-DIMENSIONAL PROJECTIVE SPACE55
Now let X be a smooth complete intersection of two quadrics in P2k with

k > 2. Most of the previous discussion extends to this case. We refer for
details to Miles thesis “Intersection of two and more quadrics” which can
be found online or to my joint preprint (AGArchive) with A. Duncan that
deals with the case when char(k) = 2.
Let Q1 , Q2 be two smooth quadrics in the pencil |L| of quadrics contain-
ing X. We know that X contains one ruling of subspaces of dimension k − 1.
Its dimension is equal to 12 k(k + 1). Since dim G(k + 1, 2k + 1) = k(k + 1),
we expect that X contains finitely many linear subspaces of dimension k.
In fact, this is always true, and the number of such subspaces (called gener-
ators) is equal to 22k . By the adjunction formula −KX = (2k − 3)η, where
η is the class of a hyperplane section. The variety X is a Fano variety of
dimension 2k − 2. Let Ak−1 (X) be the group H k−1 (X, Z) if k = C, or the
group H k−1 (X, Z` ) otherwise (or the group of algebraic cycles of dimension
k − 1 modulo rational equivalence, see [Fulton]). The classes of η k−1 and the
classes of generators span Ak−1 (X). The group Ak−1 (X) is equipped with
the intersection symmetric bilinear form
Ak−1 (X) × Ak−1 → A2k−2 (X) = Zη 2k−2 ∼

= Z.
The orthogonal complement of η k−1 is a free abelian group equipped with

a integral values symmetric bilinear form. In some basis, it coincides with
the Cartan matrix of the root system of type D2k+1 . Every generator Λ is
intersected by 2k +1 generators, and a choice of an order on this set of 2k +1
generators defines a basis in Ak−1 (X). Such a choice is called a marking of
X, and as in the two-dimensional case, we obtain that the moduli space of
marked complete intersections of two quadrics in P2k is isomorphic to the
moduli space of ordered 2k + 1 points in P1 modulo projective equivalence.
Its dimension is equal to 2k − 2.
3.6 Intersection of two quadrics in an odd-dimensional

projective space
We have already studied the intersection of two quadrics in P3 . This is a
quartic elliptic curve. Let us study such intersections in a higher-dimensional
projective space of odd dimension. We assume that X = Q1 ∩ Q2 ⊂ P2g+1
is smooth and char(k) 6= 2. By Theorem ??, the pencil |L| generated by
Q1 , Q2 contains exactly 2g + 2 singular quadrics. Their singular locus is a
point. We identify |L| with P1 and let C → P1 be the double cover ramified
over D(|L|). We can choose the equations of two quadrics V (q1 ), V (q2 ) in
|L| as in Theorem ?? and choose a basis in L formed by q1 , q2 . Then the

singular quadrics have coordinates [a1 , −1], and the double cover has the
equation
x22 − (x1 + a1 x0 ) · · · (x1 + a2g+2 x0 ) = 0 (3.6.1)
(understood to be the equation of a hypersurface in the weighted projective
space P(1, 1, g + 1)). Or, if we change the basis of L to (q1 , a2g+2 q1 − q2 ), we
may assume that the singular quadric V (a2g+2 q1 − q2 ) corresponds to the
point at infinity [1, 0], and we can write C in affine equation
y 2 − (x + a1 ) · · · (x + a2g+1 ) = 0.
This is a familiar equation of a hyperelliptic curve of genus g (a rational
curve if g = 0, an elliptic curve if g = 1).
Thus a smooth intersection of two quadrics in P2g+1 defines a hyper-
elliptic curve of genus g. Conversely, by taking such curve as in equation
(??), we can define two quadrics with equations (??). Note that the sets of
isomorphism classes of both sets are the same and coincide with the orbit of
PGL(2) on the set of distinct ordered 2g + 2 points. It is a quasi-projective
algebraic variety of dimension 2g − 1. Thus we see that the geometry of
smooth complete intersections of two quadrics in P2g+1 must be related to
the geometry of hyperelliptic curves of genus g. Of course, when g = 1, it is
not surprising.
Recall that the Jacobian variety of a nonsingular projective curve C of
genus g is an abelian variety Jac(C) whose set of k-rational points is the
group of divisor classes of degree 0 modulo linear equivalence. Over C, it is
the torus Cg /Λ, where Λ is spanned by 2g-vectors
Z Z
v i = ( ω1 , . . . , ωg ), i = 1, . . . , 2g,
γi γi
where (ω1 , . . . , ωg ) is a basis of holomorphic differential 1-forms and (γ1 , . . . , γ2g )

is a basis of homology classes of 1-cycles. The map of the group Div(C)0 of
divisor classes of degree 0 to Jac(C) isR given by Rthe Abel-Jacobi map that
c c
assigns to a point c ∈ C, the vector ( c0 ω1 , . . . , c0 ωg ) modulo the lattice.
It is defined by fixing a point c0 ∈ C. By Abel’s Theorem, the kernel of this
map is the group of principal divisors.
The following is a fundamental theorem, attributed to A. Weil.
Theorem 3.8. Let X be a smooth intersection of two quadrics in P2g+1
and C be the associated hyperelliptic curve of genus g. Then the variety
Fg−1 (X) of (g − 1)-planes contained in X is naturally isomorphic to the
Jacobian variety of C.
Proof. We will restrict ourselves only to the case g = 2, leaving the general
case to the reader. For each ` ∈ F (X) := F1 (X) consider the projection
map p` : X 0 = X \` → P3 . For any point x ∈ X not on `, the fiber over p` (x)
is equal to the intersection of the plane `x = h`, xi with X 0 . The intersection
of this plane with a quadric Q from the pencil |L| is a conic containing ` and
another line `0 . If we take two nonsingular generators of |L|, we find that
the fiber is the intersection of two lines or the whole `0 ∈ F (X) intersecting
`. In the latter case, all points on `0 \ ` belong to the same fibre. Since
all quadrics from the pencil intersect the plane h`, `0 i along the same conic
`∪`0 , there exists a unique quadric Q`0 from the pencil which contains h`, `0 i.
The plane belongs to one of the two rulings of planes on Q`0 (or a unique
family if the quadric is singular). Note that each quadric from the pencil
contains at most one plane in each ruling which contains ` (two members of
the same ruling intersect along a subspace of even codimension). Thus we
can identify the following sets:
• pairs (Q, r), where Q ∈ |L|, r is a ruling of planes in Q,
• B = {`0 ∈ F (X) : ` ∩ `0 6= ∅}.
If we identify P3 with the set of planes in P5 containing `, then the latter set
is a subset of P3 . Let D be the union of `0 ’s from B. The projection map p`
maps D to B with fibres isomorphic to `0 \ ` ∩ `0 .
Extending p` to a morphism f : X̄ → P3 , where X̄ is the blow-up of X
with center at `, we obtain that f is an isomorphism outside B and that the
fibres over points in B are isomorphic to P1 . Observe that X̄ is contained
in the blow-up P̄5 of P5 along `. The projection f is the restriction of the
projection P̄5 → P3 which is a projective bundle of relative dimension 2. The
crucial observation now is that B is isomorphic to our hyperelliptic curve
C. In fact, consider the incidence variety
X = {(Q, π) ∈ |L| × G2 (P5 ) : π ⊂ Q}.
Its projection to |L| has fiber over Q isomorphic to the rulings of planes in
Q. It consists of two connected components outside of the set of singular
quadrics and one connected component over the set of singular quadrics.
Taking the Stein factorization, we get a double cover of |L| = P1 branched
along the discriminant. It is isomorphic to C.
A general plane in P3 intersects B at d = deg B points. The preimage of
the plane under the projection p` : X 99K P3 is isomorphic to the complete
intersection of two quadrics in P4 . Taking a general hyperplane, we may
assume that the intersection of the two quadrics is nonsingular. Thus it is a

del Pezzo surface of degree 4, hence it is obtained by blowing up five points
in P2 . We know that any line in such a surface intersect five other lines.
Thus d = 5 and B is isomorphic to a genus 2 curve of degree 5 in P3 . The
exact sequence
0 → IB (2) → OP3 (2) → OB (2) → 0,
together with the fact that dim H 0 (B, OB (2)) = 9 that follows from the
Riemann-Roch formula, shows that B is contained in a unique quadric Q in
P3 . As we explained in Lecture 1, the exceptional divisor E of the blowing-up
X̃ → X is isomorphic to the product ` × P1 = P1 × P1 . It is also isomorphic
to a nonsingular quadric. The image of E under the map X̃ → P3 is a
quadric containing B. Thus B is contained in a unique nonsingular quadric
Q. It is easy to see that B must be a curve of bi-degree (2, 3). Recall that
it means that the divisor classes f1 and f2 of lines in each ruling intersect B
with degree 2 and 3. The lines from the first ruling cut out in B the linear
series of degree 2 that coincides with the canonical linear system |KB |.
We see now that the projection p` : X̄ → P3 is a birational isomorphism
which blows down the union of lines intersecting ` to the curve B. The
inverse rational map P3 99K P5 whose image is X is obtained by the linear
system of cubics containing B. In fact, cosnider the exact sequence
0 → IB (3) → OP3 (3) → OB (3) → 0.
Since deg B = 5, we get deg OB (3)) = 15. Applying Riemann-Roch, we get

h0 (OB (3)) = 14. Since h0 (OP3 (3)) = 20, we obtain that h0 (IB (3)) = 6.
Thus dim |IB (3)| = 5, and the map φB given by the linear system |IB (3)|
of cubics through B maps P3 to P5 . Every honest bisecant of B intersects a
member of the linear system at one point outside of B. Thus it is mapped
to a line contained in the image X 0 of the map. A tri-secant of B is blown
down to a point. The set of trisecants coincides with the linear series g31 on
B cut out by a ruling of the quadric containing B. So, the set of trisecants
is mapped to a line ` ⊂ X 0 . The degree of X 0 is easy to compute. A general
line in P5 corresponds to the intersection iof two cubic surfaces from |IB (3)|.
They intersect along a curve of degree 9 that contains the curve B. So, the
residual curve is of degree 4. This shows that a general line in P5 intersects
X 0 at four points, hence deg X 0 = 4. Thus the composition of rational maps
Φ = φB ◦ p` is a map X 99K X 0 . Since p + l sends a general line in X to a
secant of B and φB sends a bisecant to a line in X 0 , we see that Φ sends a
general line X to a general line in X 0 . It is easy to see that this implies that
Φ is a projective isomorphism X ∼ = X 0.
It follows from the previous discussion that X is birationally isomorphic

to the blow-up of P3 along the curve B. Since X is assumed to be smooth,
this easily implies that B is a smooth hyperelliptic curve of genus 2.
Let us construct an isomorphism between Jac(C) and F1 (X). Recall
that Jac(C) is birationally isomorphic to the symmetric square C (2) of the
curve C. The canonical map C (2) → Pic2 (C) defined by x+y 7→ [x+y] is an
isomorphism over the complement of one point represented by the canonical
class of C. Its fiber over KC is the linear system |KC | isomorphic to P1 .
Also note that Pic2 (C) is canonically identified with Jac(C) by sending a
divisor class ξ of degree 2 to the class ξ − KC .
Each line `0 skew to ` is projected to a secant line of B. In fact, h`, `0 i∩X
is a quartic curve in the plane h`, `0 i ∼
= P3 that contains two skew line com-
ponents. The residual part is the union of two skew lines m, m0 intersecting
both ` and `0 (use that the hyperplane section if a curve of arithmetic genus
one, this forces the residual part to be the union of two lines. Thus `0 is
projected to the secant line joining two points on C which are the projec-
tions of the lines m, m0 . If m = m0 , then `0 is projected to a tangent line of
B. Thus the open subset of lines in X skew to ` is mapped bijectively to an
open subset of C (2) represented by “honest” secants of C, i.e. secants which
are not 3-secants. Each line `0 ∈ F1 (X) \ {`} intersecting ` is projected to a
point b of B. The line f of the ruling of Q intersecting B with multiplicity
3 and passing through a point b ∈ B defines a positive divisor D of degree 2
such that f ∩ B = b + D. The divisor class [D] ∈ Pic2 (C) is assigned to `0 .
Finally, the line ` itself corresponds to KC . This establishes an isomorphism
between Pic2 (C) and F (X).
Remark 3.9. Note that we have shown during the proof, that X admits a
birational morphism to P3 , hence it is is a rational variety. It is one of an
examples of Fano 3-dimensional variety that happens to be rational. Not all
of them are rational. For example, a cubic hypersurface in P4 is known to
be non-rational.
Remark 3.10. Note that the proof works in any characteristic, even in
characteristic 2. In the latter case, the curve B in this case admits a sep-
arable double cover B → P1 ramified at the discriminant of the pencil. Its
equation can be given by
y 2 + a3 (u, v)y + a6 (u, v) = 0,
where a3 , a6 are binary forms of degrees 3 and 6. The zeros of a3 are the
zeros of the discriminant of the pencil. Thus a3 is given by the pfaffian
of the matrix of the bilinear form uq1 + vq2 , where V (q1 ), V (q2 ) generate
the pencil. The binary form is not uniquely defined, we can replace it
by a06 = a6 + b23 + a3 b3 , where b3 is any binary form of degree 3. One
can show that a6 can be chosen in a such a way that it vanishes at zeros
of [1, a3 ]. In this case the curve is nonsingular if and only if it has only
simple zeros at the zeros of a3 (see cited paper of U. Bhosle, Proposition
1.5. Theorem ?? gives a canonical equations of a smooth X when a3 is a
reduced polynomial of degree 3. In this case B has 3 distinct non-trivial
2-torsion divisor classes equal to [p2 − p1 ], [p3 − p1 ], [p2 + p3 − 2p1 ], where
pi = [1, ai , 0]. It is known that Jac(C)[2] is isomorphic to eitehr (Z/2Z)2 ,
or Z/2Z, or trivial. A theorem of David Leep [Journal of algebra and its
Appl. vol. 1 (2002)] states that, without assumption on a3 , X is smooth if
and only if the curve C is nonsingular. The polynomial a6 in the equation
of C is expressed in terms of the Arf-invariant of the pencil defined by
Arf = a6 /a23 .
3.7 Quadratic line complex

Recall that a smooth quadric in P5 could be identified with the Klein quadric,
the Grassmannian G = G(2, 4) = G1 (P3 ) of lines in P3 embedded via the
Plücker embedding. Thus the intersection of G with another quadric Q can
be viewed as a hypersurface in G(2, 4). It is called a quadratic line complex.
We will assume that X = G ∩ Q is smooth. We already know that X defines
a genus 2 curve, the double cover of the pencil |L| generated by quadrics
G, Q branched along the discriminant variety D(|L|). There is much more
fascinating geometry related to this correspondence.
Let
ZG = {(x, `) ∈ P3 × G}
p1 p2
v (
P3 G
be the incidence variety with its two natural projections. The fiber of the
first projection over a point x ∈ P3 is the plane πx of lines containing this
point. As we know from Lecture 1, its image under the second projection is
a plane in G. The fiber of the second projection over a line ` ∈ G, is mapped
to ` itself under the first projection. It is easy to see that these maps are
bijections, so we may identify the first fiber with a plane πx , and the second
fiber with the line `. In fact, the second projection is projective line bundle
over G, and the first projection is a projective bundle of relative dimension
3.7. QUADRATIC LINE COMPLEX 61
2. Now, let us restrict the incidence variety over X = G ∩ Q
ZX = {(x, `) ∈ P3 × X}
p1 p2
v (
P3 X
The image of the second projection is the variety F1 (X) of lines on X.

We know that it is isomorphic to Jac(C). The fiber of the first projection
over x ∈ P3 is the intersection of πx ∩ Q. It could be the whole plane, a
nonsingular conic, or a singular conic. We know that the first case does not
occur since it would imply that X contains a plane Λ. Let us see that the
smoothness assumption on X implies that this is impossible. For any point
x ∈ L the tangent hyperplanes Tx (G) and Tx (Q) are hyperplanes in P5 . By
varying x inside Λ, we define two (projective) linear maps φG : Λ → P̌5 and
φQ : Λ → P̌5 . The image of each map is the plane of hyperplanes containing
Λ. The self-map φ−1 G ◦ φQ : Λ → Λ is a projective automorphism of the
plane, hence has a fixed point. The fixed point x0 ∈ Λ corresponds to a
hyperplane which is tangent to both Q and G at the point x0 . This would
imply that X is singular at x0 . An alternative proof is to use the Lefschetz
Theorem on a hyperplane section that tells that Pic(P5 ) → Pic(X) is an
isomorphism. Thus all surfaces in X have even dimension, so X does not
contains planes.
Definition 3.11. The Kummer surface associated with a quadratic line

complex is the locus of points x ∈ P3 such that the fiber p−1
1 (x) is a reducible
conic.
Theorem 3.12. The Kummer surface is an irreducible surface K of degree

4 with 16 ordinary double points. The fibers of p1 over these points are
double lines.
Proof. We know that the variety of lines F1 (X) is isomorphic to the Jacobian
variety Jac(C). Each line in F1 (X) is a line in G, and hence it is equal to
the pencil πx,Λ = πx ∩ πΛ of lines in some plane Λ passing through a point
x ∈ Λ. The plane πx intersects Q along a reducible conic that contains this
line as its irreducible component. Thus all lines in X come in pairs, each
pair makes a reducible fiber of p1 over a point in the Kummer surface. This
shows that Jac(C) admits a degree 2 map onto K.
Recall that an abelian variety A of dimension g has the involution [−1]A
that sends a point x to −x. The fixed points of this involution are 2-torsion
points. There are 22g of them if char(k) 6= 2. The quotient space A/([−1]A )
is an algebraic variety with 22g singular points. It is called the Kummer
variety of A and is denoted by Kum(A). When g = 2, as in our case, we
obtain a surface with 16 ordinary double points. Its minimal resolution is a
K3 surface. So, in the following we are going to give a geometric construction
of Kum(Jac(C)) as a surface in P3 parameterizing pairs of lines in X = G∩Q
that lie in a plane contained in G. By definition, the pairs of these lines
form an orbit in F1 (X) = Jac(C) of some involution τ of Jac(C). Note
that any involution on an abelian variety looks like x 7→ −x + ta , where
ta is a translation map x 7→ x + a. Its fixed points satisfy 2x = a, and
form a principal homogeneous space over A[2]. In particular, their number
is 22g . By changing the origin in A, we may assume that the involution
is the negation involution. Thus, we see that our Kummer surface must
have 16 singular points. They correspond to lines ` in X such that the
plane πx is tangent to Q along `. So, we see that our Kummer surface is
isomorphic to Kum(Jac(C)). Its 16 nodes should correspond to the points
over which the fiber is a double line. In fact, by definition of the involution
on F1 ∗ X) = Jac(C), the fixed points of this involutiion correspond to pairs
of coinciding lines.
It remains to show that our Kummer surface K is a quartic surface.
To show this we have to check that a general line ` in P3 intersects k at
four points. Consider the union ∪x∈` πx of all lines that intersect `. It is
known that they form the intersection of G(2, 4) with a hyperplane Hin the
Plc̈ker space P5 that is tangent to G(2, 4) at the point represented by `. The
intersection S = H ∩ X = H ∩ G ∩ Q is a del Pezzo surface S of degree
4. The restriction of p1 to p−1
2 (S) defines a fibration S → ` whose general
fiber is a conic and singular fibers are the unions of two lines. To see that
the number of sich lines is exactly four, we use the theory of quartic del
Pezzo surface. If we exhibit S as the blow-up of 5 points, a conic bundle on
such a surface is equal to the pre-image of lines through one point. There
are 4 lines which pass through the remaining 4 points which define singular
members of the pencil.
Remark 3.13. There are many other facts about Kummer quartic surfaces
which are discussed in [DolgachevCAG] or [Griffiths-Harris]. For example,
projection from a node, gives its birational model isomorphic to the double
cover of P2 branched along the union of six lines. The six lines are all tangent
to the same conic, and the six tangency points are projectively equivalent
to to the six points in the discriminant equation of the intersection of two
3.7. QUADRATIC LINE COMPLEX 63
quadrics. Another remarkable fact is that, by considering the other ruling of

planes in the Grassmannian, one finds in P3 sixteen planes that tangent the
Kummer surface along a conic. The sixteen conics and sixteen nodes form
an abstract configuration (166 , 166 ), the Kummer configuration. One more
fact which I cannot resist not to mention is that the set of intersection points
of pairs of lines over points in the Kummer surface is a smooth surface Y in
P5 of degree 8 birationally isomorphic to the Kummer surface. IfPwe write
equation of X as the intersection of two quadrics with equations 6i=1 t2i =
P 6 2
P6i=1 a2i ti2, then the equation of Y is given
2
by adding one more equation
i=1 ai ti = 0. The six points [1, ai , ai ] the plane are the images of the
points in the discriminant variety of X under the Veronese map. They
correspond to the Weierstrass points of the genus 2 curve C.
Remark 3.14. A conic bundle is a flat morphism of smooth varieties f :
X → Y that whose general fiber is isomorphic to P2 and singular fibers are
isomorphic to the union of two P1 ’s intersecting transversally at one point.
The subvariety of Y parameterizing singular fibers is called the discriminant
of a conic bundle. . In our previous discussion we found an example of a
conic bundle ZX → P3 whose discriminant is a Kummer surface. the variety
ZX is isomorphic to a P1 -bundle over X = G(2, 4) ∩ Q. Since the latter is
rational, the total space ZX of the conic bundle is rational. In general, it is
avery difficult problem to decide whether the total space of a conic bundle
is rational even when its base is rational. We will see in a future lecture,
and example of Artin and Mumford when this is not the case.
sectionPrincipally polarized abelian varieties Here we give a more ab-
stract construction of the Kummer surface of the jacobian variety as a 16-
nodal quartic surface in P3 . Let A be an abelian variety of dimension g. A
principal polarization on A is an ample divisor class ∆ such that h0 (∆) :=
dim H 0 (A, OA (∆)) = 1. By Riemann-Roch on an abelian variety, we have
g
h0 (m∆) = mg! ∆g . Thus ∆ is a principal polarization if and only if ∆g = g!.
Taking m = 2, we obtain a linear system |2∆| with dim |2∆| = 2g − 1.
Let ta denote the translation automorphism of A by a point a ∈ A. One
can find a such that [−1]∗A (t∗a (∆)) = t∗a (∆). Replacing ∆ with t∗a (∆), we
find a principal polarization such that [−1]∗ (∆) = ∆. It is called a sym-
metric principal polarization. The map given Φ∆ : A → |∆|∗ ∼
g
= P2 −1
given by the linear system |2∆| commutes with the involution [1]A and
hence factors through the Kum(A) = A/([−1]A ). One can show that it
g
defines an embedding of Kum(A) in P2 −1 unless (A, ∆) is a reducible po-
larization, i.e. (A, ∆) ∼= (A1 , ∆1 ) × · · · (A1 , ∆1 ). The latter means that
A∼ = A1 × · · · Ak and ∆ = p∗1 (∆1 ) + · · · p∗k (∆k ). It follows that, if this is not
g −1
the case, Kum(A)0 = φ2∆ (Kum(A)) is a g-dimensional subvariety of P2
of degree
deg(Kum(A)) = (2∆)g /2 = 2g−1 g!,
In particular, we seee that the image is quartic surface if g = 2. In this case
the polarization is irreducible if and only A ∼ = Jac(C) fotr some smooth curve
C of genus 2. The images of the set of 2-torsion points is a set of singular
points on φ(Kum(A)). The number of them is equal to 22g if char(k) 6= 2.
Each singular point is formally isomorphic to the singular point of the cone
over a Veronese variety νg (Pg−1 ) in Pg .
It follows from the definition of the map φ2∆ defined by the linear system
|2∆ that if we choose a representative 2D of 2∆ with [−1]∗A (D) = D, then,
the image of 2D is cut out in Kum(A)0 by a hyperplane that is tangent to
Kum(A)0 along the image of D. There are 2g such divisors, each is obtained
from one of them by translation ta , where a ∈ A[2] is a 2-torsion point. Thus
we have 22g hyperplanes that are tangent to Kum(A)0 along the divisor D0 ,
the image of D with respect to φ2∆ . These hyperplanes are called tropes.
So, there are 22g tropes. One can show that each trope contains 2g−1 (2g − 1)
singular points, and each singular point is contained in so many tropes. This
realizes an abstract configuration (22g , 22g
2g−1 (2g −1) 2g−1 (2g −1)
) which is called the
Kummer configuration.
In our case when A = Jac(C) and g = 2, we see that the configuration
is (166 , 166 ). In this case the principal polarization is defined by the image
of the Abel-Jacobi map C → Jac(C), c 7→ [c − c0 ]. The divisors D are
isomorphic to C, the image of C is a conic D0 in the Kummer quartic surface
that contains 6 singular points. They are the branch points of the double
cover D → D0 .
Note that the construction works also if the characteristic is equal to 2.
In this case, we have A[2] ∼ = (Z/2Z)g−k , where 0 ≤ k ≤ g. For example,
in our situation, we still have a quartic Kummer surface, however, it has
4, 2 or 1 singular points. The singular points are more complicated than
ordinary nodes. For example, in the last case, the Kummer surface is a
rational surface with one (elliptic) singularity.
Lecture 4
Conic bundles
4.1 The Brauer group

Recall the Brauer group of a field k is the group of equivalence classes of
central simple algebras over k. This means that the algebra has center
equal to k and it has no two-sided ideals. Each algebra is isomorphic to
the matrix algebra Mn (K) over some separable extension K of k. This
implies that dimk A = n2 . Another way to see this is the fact that any
central simple algebra is isomorphic to the matrix algebra over a division
ring (a non-commutative associative algebra over a field where each non-
zero element is invertible). The equivalence for simple algebras is defined by
A1 ≡ A2 if A ⊗ Mk (k) ∼= A2 ⊗ Mn (k) for some k, n. A simple central algebra
is a k-form of the matrix algebra Mn (k), hence it defines an element of the
Galois cohomology H 1 (k, PGLn ) (we use that, by Skolem-Noether theorem,
the group of automorphisms of Mn (k) is isomorphic to PGLn ). The exact
sequence of algebraic groups
1 → Gm → GLn → PGLn → 1
defines, via the coboundary map, the class in Br(k) := H 2 (k, Gm ), the
Brauer class. By Hilbert 90 Theorem, H 1 (k, GLn ) = {1}, hence H 1 (k, PGLn )
becomes a subgroup of the group Br(k). One can show that, in fact, it co-
incides with this group. The group law is defined by the tensor product of
algebras.
Example 4.1. Let A be the quaternion algebra (a, b)k over k. It is generated
over k by elements i, j such that i2 = a, j 2 = b, ij = −ji, where a, b ∈ k∗ .
When k = R and a = b = −1, we get the usual definition of the quaternion
65
66 LECTURE 4. CONIC BUNDLES
algebra (where k = ij). It is easy to see that (au2 , b)k ∼= (a, b)k , thus if a or
∼
b is a square in k, we have (a, b)k = (1, 1)k (by sending
1 7→ ( 10 01 ) , i → 10 −1
0 , j 7→ 0 1 0 −1

−1 0 , k 7→ 1 0 .)
√ √
Let K = k( a or K = b). Then AK ∼ = M2 (K). It is clear that
A ⊗k A is generated by x ⊗ y, x, y = 1, i, j, k and (x ⊗ y)2 is a square in k.
Thus it becomes isomorphic to the matrix algebra over k. This shows that
the Brauer class of a quaternion algebra is of order 2. It is a still an open
problem where any element of order 2 in Br(k) is equivalent to a quaternion
algebra.
Now let us move from 0-dimensional scheme Spec k to any scheme S. We
use étale topology of S that replaces the category of separable extensions in
the Galois theory of fields. An immediate generalization of a central simple
algebra is the notion of an Azumaya algebra. It is a sheaf of Algebras over
S locally isomorphic in étale topology to the algebra E\d(E) of some locally
free sheaf E over S. Two Azymaya algebras A, B are called equivalent if
A ⊗ E\d(E1 ) ∼= A2 ⊗ E\d(E2 ) for some locally free sheaves E1 , E2 . We have
an exact sequence of sheaves on S
1 → Gm,S → GLn,S → PGLn,S → 1
and the exact sequence of cohomology gives a homomorphism
φ : H 1 (S, PGLn,S ) → H 2 (S, Gm ).
In general, this homomorphism is neither injective nor surjective. The group

H 1 (S, PGLn,S ) is the Brauer group of S, and the group H 2 (S, Gm ) is called
the cohomological Brauer group. However, under some conditions on S, for
example, when S is a smooth algebraic variety, the two groups are isomorphic
and denoted by Br(X).
Another geometric interpretation of a cohomology class from H 1 (S, PGLn,S )
is an isomorphism class of a Severi-Brauer variety. It is a variety X over
S such that the fibers of the structure morphism X → S are isomorphic
to a projective space Pr . Locally, in étale topology, it is isomorphic to a
projective bundle over S, i.e Proj S(E) for some locally free sheaf of rank
r + 1 over S.
For example, the Severi-Brauer variety whose cohomology class coincides
with the class of a quaternion algebra Q(a, b) is a conic over k with equation
C : ax2 + by 2 − z 2 = 0. If we take the intersection of this conic with a line
√
x = 0, we obtain a point in C(K), where K is a quadratic extension k( a).
4.1. THE BRAUER GROUP 67
One can show that the group of endomorphisms of this quadratic algebra
over k is isomorphic to the quaternion algebra (a, b)k .
Globalizing this example, we obtain that a Severi-Brauer variety over S
of relative dimension 1 is given by a smooth conic bundle over S.
The latter is given by a locally free sheaf of rank 3 over S and a section
of q of S 2 E. It corresponds to a section of OP(E) (2) under the isomorphism
π∗ OP(E) (2) ∼= S 2 E. If we view q as a linear map E ∨ → E, its rank at each
point is equal to 3, so in each fiber of P(E) → S, its defines a nonsingular
conic.
Consider the Kummer exact sequence of sheaves in étale topology:
[n]
1 → µn → Gm → Gm → 0 → 0.
We use that Pic(S) ∼= H 1 (S, Gm ). The exact sequence gives an exact se-
quence of group cohomology:
0 → Pic(S) ⊗ Q/Z → H 2 (S, µn ) → Br(S)[n] → 0,
where Br(S)[n] is the group of n-torsion elements. This also gives an exact
sequence
0 → Pic(S) ⊗ Q/Z → H 2 (S, Q/Z) → Tors(Br(S)) → 0, (4.1.1)
Now we use the exact sequence
0 → Z → Q → Q/Z
to get an exact sequence
0 → H 2 (S, Z) ⊗ Q/Z → H 2 (S, Q/Z) → Tors(H 3 (S, Z) → 0.
Another way to see this exact sequence is to consider the exact sequence
0 → Z → Z → Z/nZ → 0
that gives an exact sequence
H 2 (S, Z)/nH 2 (S, Z) → H 2 (S, Z/nZ) → H 2 (3, Z)[n] → 0,
and then take the inductive limit. Together with (??), we get a commutative
diagram
0 / Pic(S) ⊗ Q/Z / H 2 (S, Q/Z) / Br(S) /0
c1

0 / H 2 (S, Z) ⊗ Q/Z / H 2 (S, Q/Z) / Tors(H 3 (S, Z) /0
This shows that the homomorphism Br(S) → Tors(H 3 (S, Z) is always sur-
jective. Also, if the first Chern class map c1 : Pic(S) → H 2 (S, Z) is an
isomorphism (e.g. S is a rational variety), we have an isomorphism
Br(S) ∼
= Tors(H 3 (S, Z).
Here we used cohomology with integer coefficients, this requires the as-
sumption that k = C. However, the same argument, replacing Z with Zl ,
l 6= p = char(k). Then we obtain isomorphism
Br(S)(l)6=p ∼
= Tors(H 3 (S, Zl ),
where Br(s)6=p is the Brauer group modulo p-torsion.
Proposition 4.2. Let f : X 0 99K X be a birational isomorphism of smooth

varieties. Then
Tors(H 3 (X, Z) ∼
= Tors(H 3 (X 0 , Z).
Proof. Since any birational morphism of nonsingular varieties is decomposed

into blow-ups and blow-downs with non-singular centers, it is enough to as-
sume that f : X 0 → X is a blow-up with a nonsingular center Y . Let
f ∗ : H k (X, Z) → H k (X 0 , Z) be the usual pull-back homomorphism of co-
homology and f∗ : H k (X 0 , Z) → H k (X, Z) be the homomorphism obtained
by identifying, via the Poincaré Duality, H k (X 0 , Z) with Hdim X 0 −k (X 0 , Z)∨ .
Since f is birational, f∗ f ∗ [X] = [X], hence, by the projection formula,
f∗ (x · f ∗ (y)) = f∗ (x) · y shows that f∗ f ∗ is the identity. Thus H k (X, Z) in-
jects in H k (X 0 , Z). In particular, Tors(H 3 (X, Z)) injects in Tors(H 3 (X 0 , Z).
Next we use the Leray spectral sequence for f . Since the fibers over points
in the center Y of the blow up are projective space, we obtain Rk f∗ Z = 0
for odd k. Consider the Leray spectral sequence
E p,q := H p (X, Rq f∗ Z) ⇒ H ∗ (X 0 , Z)
The boundary homomorphism d2 : E p,q → E p+2,q−1 all zeroes. Thus the

spectral sequence degenerates and gives an exact sequence
0 → E 3,0 → H 3 (X 0 , Z) → E 1,2 → 0.
Since E 1,2 = H 1 (Y, Z) is torsion-free (this follows from the formula for
universal coefficients) and we know that from above that E 3,0 = H 3 (X, Z)
is a direct summand of H 3 (X 0 , Z), we get that the torsion of two groups
coincide.
4.2. CAYLEY QUARTIC SYMMETROID 69
Note that more generally, Grothendieck proves that Br(X) is a birational

invariant.
Corollary 4.3. Let X be a nonsingular rational variety, then
Br(X) = 0.
We use that c1 : Pic(X) → H 2 (X, Z) is always an isomorphism for a

rational variety. Thus Br(X) ∼
= Tors(H 3 (X, Z)).
4.2 Cayley quartic symmetroid

Let L be a web of quadrics in P3 . We assume that L is general in the sense
that it intersects D3 transversally. In particular, S = D(|L|) is a quartic
surface with 10 ordinary double points. It is called a Cayley quartic sym-
metroid. Note that it is not true that any quartic surface with 10 nodes is
isomorphic to a Cayley quartic symmetroid, although the number of param-
eters for projective equivalence classes for quartics with 10 nodes and Cayley
symmetroids is the same and is equal to 9. A Cayley quartic symmetroid
is characterized by the property that the projection from a node exhibits it
as a double cover branched along the union of two cubic curves such that
there is a conic that is tangent to all of them at three points. Recall that, if
we choose the node to have coordinates [1, 0, 0, 0], the equation of a quartic
surface with this node becomes
x20 Q2 (x1 , x2 , x3 ) + 2x0 F3 (x1 , x2 , x3 ) + G4 (x1 , x2 , x3 ) = 0.
The branch curve of this cover is given by the equation
B : F32 − Q2 G4 = 0.
It is a curve of degree 6 such that the conic V (Q2 ) is tangent to it at six

points V (F3 ) ∩ V (Q2 ). Since a quartic with 10 nodes has 9 other nodes
besides the center of the projection, we see that B has 9 singular points. In
our case B is the union of two cubics intersecting at 9 points. We will prove
it later. Also, we will explain the relationship between the Cayley quar-
tic symmetroid and Enriques surfaces. For a non-believer of this strange
fact, we can recompute the count of constants by other method. As double
covers, quartic symmetroids are parameterized by webs of quadrics mod-
ulo projective linear group PGL(4). The webs are parameterized by the
Grassmannian G3 (|OP3 (2)|) of dimension 4(10 − 4) = 24. The number of
parameters is again 24 − 15 = 9. Also, we can recompute the number of

parameters of 10-nodal quartics by counting the number of parameters of
9-nodal plane sextics that admit an everywhere tangent conic. The number
of them is equal dim |OP2 (6)| − 1 − 9 − 8 = 27 − 18 = 9.
Note that the moduli space of Cayley quartic symmetroids is of dimen-
sion 9. In fact, pairs of cubics depend on 18 parameters (since dim |OP2 (3)| =
9) and the preojective linear group PGL(3) is of dimension 9. On the other
hand 10-nodal quartic surfaces in P3 also depend on 9 parameters. In fact,
dim |OP3 (4)| = 34, each node gives one condition and dim PGL(4) = 15, all
of this gives again 9 = 34 − 10 − 15 parameters. This shows that quartic
symmetroids defines an irreducible component in the space of all 10-nodal
quartic surfaces.
4.3 Artin-Mumford counter-example to the Lüroth

Problem
Recall that a variety X is unirational (resp. rational ) if there exists a
dominant rational map PN 99K X (resp. a birational map). The Lüroth
Problem asked whether a unirational algebraic variety over an algebraically
closed field of characteristic 0 is always rational. This is always true when
dim X ≤ 2 but not always true in higher dimension. The first (rogorous)
counter-examples were forund in 1972 by three different teams M. Artin and
d. Mumford, H. Clemems and Ph. Griffiths, V. Iskovskilh and Yu. Manin.
Here, following A. Beauville, we give an application of our study of linear
systems of quadrics to discuss Artin-Mumford example.
Proposition 4.4. Let X → P3 be the double cover of P3 branched along the

Cayley quartic symmetroid S = |L|. Then X is unirational.
Proof. Let X = {(Q, `) ∈ |L| × G1 (P3 )}. Since a nonsingular quadric in

P3 contains two ruling of lines, a general fiber of the first projection to |L|
has two connected components isomorphci to P1 . By Stein factorization, it
factors through the double cover of |L| branched along D(|L|). This doubel
cover can be identified with X. Let ` be a general line in P3 . Then a quadric
from |L| contains ` if and only if it contains 3 distinct points on it. This gives
three conditions in order that Q contains `. Since dim |L| = 3, we obtain
that there is a unique quadric containing `. Thus the second projection is
of degree 1, hence X is birationally isomorphic to G1 (P3 ), and, hence, it is
rational. Since X is the image of X under a regular map, is unirational.
4.3. ARTIN-MUMFORD COUNTER-EXAMPLE TO THE LÜROTH PROBLEM71
Remark 4.5. In fact, one can show that a double cover of Pn branched along
an irreducible quartic hypersurface is unirational (see [Beauville, Lüroth
Problem]).
The next theorem belongs to M. Artin and D. Mumford. We follow

Beauville’s proof.
Theorem 4.6. X is not rational.
Proof. It follows from the proof of the previous theorem that the projection
p1 : X → X has fibers isomorphic to P1 . Under the Plücker embedding,
they are conics. Let X o be the smooth locus of X. Its complement consists
of 10 isolated points. The restriction X o → X o of of p1 over X o is a Severi-
Brauer variety that defines, if it is not trivial, an element in Br(X o ) of order
2. Suppose it is trivial. Then it is a vector bundle over X o , and hence
admits a rational section. So, it suffices to show that it does not admit such
a section.
Suppose σ : X o 99K X o is such a section. Let (Q, r), (Q, r1 ) be two
points in X lying in π −1 (Q), Q ∈ |L|. Here r1 , r2 are two rulings of lines
in Q. Then σ map (Q, r1 ), (Q, r2 ) to two points to two lines `1 ⊂ r1 and
`2 ⊂ r2 in Q. These lines intersect at one point x = `1 ∩ `2 . Thus, σ defines
a rational section |L| → Q, where Q = {(x, Q) ∈ P3 × |L| : x ∈ Q} is the
universal family of the web of quadrics.
Let us show that the universal family F of any base-point-free r-dimensional
linear system L| of hypersurfaces of degree d in Pn has no rational sec-
tions. Suppose it has a rational section σ : |L| → Q. Then its closure
is a subvariety of F birationally isomorphic to |L|. Let η ∈ H 2r (Q, Z)
be its cohomology class. The subvariety F is a hypersurface in |L| × Pn
given by an equation of bi-degree (1, d). Since |L| is base-point-free, r ≥ n
and dim F = r + n − 1 > 2n − 2. By Lefschetz’s Theorem on a hyper-
plane section, H 2n−2 (|L| × Pn ) → H 2n−2 (F, Z) is an isomorphism. Thus,
H 2n−2 (F, Z) is spanned by the images of the classes hi1 ·h2n−1−i , where h1 , h2
are the pre-images of the classes of hyperplanes in |L| and Pn . Under the fist
projection q : F → |L|, the image of hi1 · h2n−1−i is equal to hi1 · p∗ (hn−1−i
2 ).
Since dim |L| > n − 1, they are all zero, except when i = 0, in which case
q : h2n−1 → |L| is of degree d. hence, the image of q∗ to H 0 (|L|, Z) ∼ = Z is
0
equal to dZ. But the image of q∗ (η) must be a generator of H (|L|, Z). This
contradiction proves the theorem.
Thus we have seen that the P1 -fibration over X o is not isomorphic to a
P1 -bundle, hence gives a non-zero 2-torsion element in Br(X o ). Let X̃ be the
blow-up of P3 at the nodes of X. Its exceptional divisors over the singular
points pi are isomorphic to quadrics Ei . In the commutative diagram

c1
Pic(X̃) / H 2 (X̃, Z)
r

c1
Pic(X 0 ) / H 2 (X o , Z)
the top horizontal arrow is surjective. To see this use that H 2 (X̃, OX̃ ) = 0
since X̃ is a unirational and apply the exponential exact sequence
∗
0 →→ Z → OX̃ → OX̃ → 0.
Since X̃ \ X is the disjoint union Y of 10 quadrics, the Gysin exact se-

quence H 2 (X̃, Z) → H 2 (X o , Z) → H 1 (Y, Z) = 0 shows that the restric-
tion homomorphism r is surjective. Thus, by Proposition ??, we get that
Tors(H 3 (X o , Z)) 6= 0. Using again the Gysin exact sequence
0 → H 3 (X̃, Z) → H 3 (X 0 , Z) → H 2 (Y, Z)
we find that Tors(H 3 (X o , Z)) ∼
= Tors(H 3 (X̃, Z)). This proves that X̃ and
hence X is not rational.
Remark 4.7. Suppose |L| has a base point. Then, the universal family
F → |L| has a section. It assigns to the quadric Q ∈ |L| the one of the
base points x0 of |L|. Thus, the conic bundle X → X acquires a section
too. It assigns to (Q, r) the line ` ∈ r that contains the point x0 . Thus,
the Severi-Brauer variety is trivial and X becomes stably rational (i.e. the
product with P1 becomes a rational variety). Note that the non-vanishing
of the Brauer group implies that the variety is not stably rational.
An example of the case when the web |L| acquires a base point is the web
of quadrics passing through 6 points in a general position in P3 . The discrim-
inant hypersurface in this case is the Kummer quartic surface Kum(Jac(C)),
where the curve C is the double cover of the unique rational normal curve
passing through the six points ramified over these points. The six base
points are additional nodes of the quartic surface, so all together we have
sixteen of them. The double cover X of P3 branched over the Kummer sur-
face is a rational variety. To see this we consider the Segre 10-nodal cubic
hypersurface in P4 It is projectively isomorphic to the cubic hypersurface
given in P5 by two equations (one of them is linear)
6
X 6
X
x3i = xi = 0.
i=1 i=1
4.3. ARTIN-MUMFORD COUNTER-EXAMPLE TO THE LÜROTH PROBLEM73
The group S6 acts on it by permutation of coordinates. Tt contains 10

singular points forming the S6 -orbit of the point [1, 1, 1, −1, −1, −1]. It also
contains 15 planes forming the S6 -orbit of the plane x1 + x2 = x3 + x4 = 0.
It is known that the Segre cubic is isomorphic to the image of P3 under a
birational map given by the linear system of quadrics through 5 points (see
[Dolgachev]). In particular, it is a rational variety. Now, the projection from
a nonsingular point defined a birational isomorphism to the double cover of
P3 branched along a Kummer quartic surface. Ten of the 16 nodes of the
Kummer surface are the projections of the ten nodes of the Segre cubci.
The other 6 nodes are the images of the six lines that pass through any
nonsingular point on the Segre cubic.
Remark 4.8. Let f : X ⊂ P4 be a nonsingular cubic hypersurface and `

is a line on it (lines on X are parameterized by a surface, called the Fano
surface of lines on X). Let X 0 → X be the blow-up of `. Let f : X 0 → P2
be the projection morphism with center at `. For any point x 6 `, the plane
spanned by x and ` intersects X along the union of ` and a conic. The conic
is isomorphic to the fiber f −1 (f (x)). Let C be the plane curve parametrizing
reducible fibers. One can show that C is a plane curve of degree 5 and the
fibers of its points are isomorphic to reduced reducible conics. This defines
a non-ramified double cover C̃ → C whose fibers are components of these
reducible conics. One of the proofs of non-rationality of X (it is always is
unirational) consists of proving that the Prym variety
Prym(C̃/C) : Ker(N m : Jac(C̃) → Jac(C))
is isomorphic to the intermediate Jacobian variety J(X) and it is not iso-

morphic, as a principally polarized abelian variety to a Jacobian variety. It
would be interesting to find the proof of non-rationality by showing that the
conic bundle over P2 \ C is a non-trivial Severi-Brauer variety.
Lecture 5
Quartic symmetroid and

Enriques surfaces.
Let us discuss the Cayley quartic symmetroids in more details. Let L ⊂

E ∼ = k4 be generated by four quadrics Qi = V (qi ), i = 1, 2, 3, 4. If L is
general, the base scheme Bs(|L|) = Q1 ∩ Q2 ∩ Q3 ∩ Q4 is empty (too many
homogeneous equations in 4 variables). Let bi = bqi . Being a symmetric
bilinear form in E, it defines a section si of an invertible sheaf
OP3 ×P3 (1, 1) := p∗1 OP3 (1) ⊗ p∗2 OP3 (1) ∼

= OP3 ×P3 (1),
where P3 × P3 is considered to the Severi variety S3,3 be in P15 by the Segre

embedding P3 ×P3 ,→ P15. Let P B(|L|) be the intersection Z(s1 )∩· · ·∩Z(s4 )
of the schemes of zeros of these sections. This is a complete intersection of
four divisors of type (1, 1) in P3 × P3 , or as four hyperplane sections of the
Severi variety. Since ωP3 ×P3 ∼= OS3,3 (−4), by the adjunction formula,
ωPB(|L| ∼
= OPB(|L|) .
Thus, PB(|L|) is a K3 surface, assuming that it is nonsingular. Since the

bilinear forms bi are symmetric, the subvariety PB(|L|) is invariant with
respect to the involution σ defined by switching the factors. Its set of
fixed point is the diagonal. A fixed point (x, x) = ([v], [v]) satisfies qi (v) =
bi (v, v) = 0, i = 1, 2, 3, 4, hence belongs to the base scheme. By assumption,
it is empty, hence σ is fixed-point-free. The quotient S = PB(|L|)/(σ) is an
Enriques surface and the surface PB(|L|) is its K3-cover.
Consider a morphism PB(|L|) → G1 (P3 ) that assigns to a point (x, y) ∈
PB(|L|) the line hx, yi spanned by the points x, y. Since PB(|L|) does not
intersect the diagonal, the map is well-defined and it factors through S.
75
76LECTURE 5. QUARTIC SYMMETROID AND ENRIQUES SURFACES.
Suppose hx, yi = hx0 , y 0 i. Let x = [v], y = [u], x0 = [v 0 ], y 0 = [u0 ], we can

write v 0 = αu + βv, u0 = α0 u + β 0 v to obtain
0 = bi (u0 , v 0 ) = αα0 bi (u, u) + ββ 0 bi (v, v) + (αβ 0 + α0 β)bi (u, v)
= αα0 bi (u, u) + ββ 0 bi (v, v) = qi (λu) + qi (µv),
where λ2 = αα0 , µ2 = ββ 0 . This gives
0 = bi (λu, µv) = qi (λu) + qi (µv) − qi (λu + µv) = −qi (λu + µv) = 0.
This shows that the point [λu + µv] belongs to Bs(|L|) contradicting the
assumption
Let ` = h[v], [u]i, where ([v], [u]) ∈ PB(|L|). A quadric Q = V (q) ∈ |L|
contains ` if and only if it contains 3 points on `. Since 0 = bq (u, v) =
q(u) + q(v) − q(u + v), if Q contains the points [v], [u], it automatically
contains the third point [u + v] on `. This implies that there will be a pencil
of quadrics in |L| that contains ` but just a unique quadric containing ` if it
were a general line. Conversely, if ` is contained in a pencil of quadrics from
|L| the previous equality implies that bq (u, v) = 0 for all q ∈ L, and hence
` = h[v], [u]i, where ([v], [u]) ∈ PB(|L|)
A line in a web |L| that is contained in a pencil of quadrics form |L| is
called a Reye line. Thus we see that the image of the map PB(|L|) → G1 (|E|)
is a surface parameterizing Reye lines of |L|.
A surface in G1 (P3 ) is called a congruence of lines. We know that
H 2 (G1 (P3 ), Z) = Zσ,

H 4 (G1 (P3 ), Z) = Zσ1 + Zσ2 ,
where σ is the class of a hyperplane section in the Plücker embedding

G1 (P3 ) ,→ P5 , and σ1 , σ2 are the classes of planes in of two rulings of planes
in a 4-dimensional quadric G1 (P3 ). We may assume that σ1 = [πx ], σ2 =
[πΛ ]. We use the intersection theory on the quadric G1 (P3 ). We have
σ12 = [πx ∩ πy ] = [point], and σ22 == [πΛ ∩ πΛ ] = [point]. Thus σi2 = 1.
Also, we have σ1 · σ2 = [πx ∩ πΛ ] = [∅], so σ1 · σ2 = 0. We can represent σ
by the variety π` of lines intersecting a fixed line `. We have π` ∩ π`0 ∩ πΛ
is the unique line in Λ intersecting the points ` ∩ Λ and `0 ∩ Λ. This gives
σ 2 · σ1 = 1. Similarly we get σ 2 · σ2 = 1. Thus σ 2 = σ1 + σ2 . All of this is a
special case of the Schubert calculus that describes the intersection theory
of Grassmannians (see [Fulton]).
77
Let Z ⊂ G1 (P3 ) be a congruence of lines in P3 and let [Z] ∈ H 4 (G1 (P3 ), Z)

be its cohomology class. Then [Z] = mσ1 + nσ2 and intersecting with σ1 , σ2 ,
and using the previous computations, we find that
m = [Z] · σ1 = #{lines in Z that contain a general point in P3 },

n = [Z] · σ2 = #{lines in general plane in P3 that are contained in Z}.
The number m (resp. n) is called the order (resp. class) of the congruence
Z, The pair (m, n) is called the bidegree of Z. Note that the degree of Z in
the Plücker embedding is equal to
deg Z = σ 2 · [Z] = σ 2 · (mσ1 + nσ2 ) = m + n.
Let Z be the congruence of Reye lines of |L|, called the Reye congruence
of |L|.
Proposition 5.1. The bidegree of the Reye congruence of |L| is equal to

(7, 3).
Proof. Fix two general planes |N1 | and |N2 in |L|. Suppose ` is contained
in a pencil P of quadrics in |L| which is not contained in |N1 | or |N2 |. Since
P intersects |M1 | and |M2 | at one point, the line ` is contained in unique
quadric in |N1 and in a unique quadric in |N2 |. Let X be the variety of lines
contained in some quadric from a net of quadrics in P3 . It is a hypersurface
in G1 (P3 ), called the Montesano cubic complex. Let us see that its class
in H 2 (G1 (P3 ), Z) is equal to 3σ, i.e. it is equal to the intersection of the
quadric G1 (P3 ) and a cubic hypersurface. It is an example of a Fano variety
of degree 6 in P5 . To see this, it is enough to compute the number of lines in
X that are contained in πx ∩πΛ for some general plane Λ and a general point
x ∈ Λ. When we restrict the net |M | to the plane Λ we get a net N (Λ) of
conics in Λ ∼= P2 . Its discriminant curve is a cubic curve. It parameterizes
reducible conics in the net. Let P(Λ, x) be the pencil of conics in N (Λ)
that pass through a general point x ∈ Λ. Then its discriminant consists of
3 points, and thus there are three singular conics passing through x. When
x is general enough we may assume that x is not the singular point of the
conic. Thus there will three line components passing through x, and hence
deg X = 3.
Thus, we see that our congruence is contained in the intersection of two
Montesano complexes. The latter is a surface of class (3σ)2 = 9(σ1 + σ2 ).
So, its bidegree is equal to (9, 9). It remains to compute the bidegree of the
residual surface and to show that it is equal to (2, 6). The residual surface
is the surface of lines that is contained in some quadric from the pencil
|M1 | ∩ |M2 |. The base curve of this pencil is a quartic elliptic curve. As a
curve on a nonsingular quadric, it is a curve of bidegree (2, 2). Thus any
line contained in some quadric will intersect this curve at 2 points. This
we have to compute the bidegree of the congruence of bisecant lines of a
quartic elliptic curve. The number of secants passing through a general
point is equal to 2 because, projecting from this point we get a plane curve
of degree 4 of arithmetic genbus 3 and geometric genus 1, so it must have
two nodes. Thus the order of the congruence is equal to 2. Now intersecting
the quartic curve with a general plane we get four points. Thus the number
of bisecants is equal to 42 = 6. So, the class of the residual congruence is

equal to (2, 6), as asserted.
Theorem 5.2. Assume that D(|L|) does not contain lines. The K3-cover
PB(|L|) of the Reye congruence Enriques surface S is a minimal resolution
of the Cayley quartic symmetroid. It is also isomorphic to a nonsingular
quartic surface in P3 with 10 lines equal to the singular loci of quadrics of
corank 2 in the web.
Proof. Let D(|L|) be the quartic symmetroid of a general web of quadrics

|L| in P3 . Let D̃(|L|) = {(Q, x) ∈ |L| × P3 : x ∈ Sing(Q)} be the incidence
variety which we had already used in discussing the discriminant variety of
a linear system of quadrics. For the complete linear system of quadrics, the
first projection defines a resolution of singularities of the discriminant variety
D3 of quadrics in P3 . One can show that, when |L| intersects transversally
D3 , the projection p1 : D̃(|L|) → D(|L|) is still a resolution of singularities.
Its fibers over each of the ten singular points is isomorphic to the singular
line of the corresponding quadric of corank 2. This is (−2)-curve on the
K3 surface D̃(|L|). Now let us consider a rational map st : D(|L|) 99K P3
(the Steiner map) that assigns to a singular quadric Q ∈ D(|L|) its singular
point. The map is not defined at the singular points of D(|L|) but extends to
a regular map of D̃(|L|). Let us find the image St(|L|) of the map. Choose
a basis (q1 , q2 , q3 , q4 ) of L and let (A1 , A2 , A3 , A4 ) be the corresponding ma-
trices of the bilinear forms P bqi ’s. Let x = [v] = [x1 , x2 , x3 , x4 ] be the singular
point of a quadric V ( λi qi ) of corank 1. It spans the null space of the ma-
P (1) (4)
trix λi Ai . Let us write Ai in terms of its columns Ai = [Ai , . . . , Ai ].
Then
4 4 4
(j)
X X X
( λ i Ai ) · v = λi ( xj Ai ).
i=1 i=1 j=1
79
Considered as a system of linear equations in unknowns λ1 , . . . , λ4 , it has a

non-trivial solution if and only if
4 4 4 4
(j) (j) (j) (j)
X X X X
det[ x j A1 , x j A2 , x j A3 , xj A4 ] = 0.
j=1 j=1 j=1 j=1
Obviously, it is a homogeneous polynomial of degree 4 in the coordinates

x1 , . . . , x4 . Thus, we see that D(|L|) is birationally isomorphic to the quartic
surface St(|L|) and the Steiner map extends to an isomorphism D̃(|L|) →
St(|L|). Thus St(|L|) is isomorphic to a minimal resolution of the Cayley
quartic symmetroid.
It remains to prove that PB(|L|) is also isomorphic to the surface St(|L|).
For this we consider the first projection PB(|L|) → P3 and show that its
image coincides with St(|L|). Let ([v], [u]) ∈ PB(|L|), then the hyperplane
bi (v) = 0 is the polar of Qi at the point [v]. The intersection of these four
polar hyperplanes contains the point y = [u]. Thus the linear forms P bi (v) are
linear dependent, hence there is a non-trivial
P4 linear relation λ i i (v) = 0.
b
This shows that the quadric Q = V ( i=1 λi qi ) is singular at x = [v]. So
the point [v] belongs to St(|L|). It is clear that y ∈ p−1 1 (x). Suppose
that y 0 ∈ p−1 1 (x) then the previous argument shows that the intersection of
0
polar hyperplanes bi (v) = 0 contains the line hy, y i. Hence there is at least
one-dimensional linear space of linear relations between the hyperplanes,
i.e. there is a pencil of quadrics with a singular point at x = [v]. This
implies that D(|L|) contains a line. Thus, under our assumption, the map
PB(|L|) → St(|L|) is bijective. Now the fiber of the st ˜ : D̃(|L|) → St(|L|)
over the point x is the same space of quadrics singular at x. Thus, under
our assumption, the surfaces D̃(|L|), St(|L|), PB(|L|) are isomorphic.
Remark 5.3. Let PB(|L|) ⊂ |E| × |E| ,→ |E ⊗ E| ∼ = P14 be the Segre
embedding. Since PB(|L|) is equal to the intersection of 4 hyperplanes, it is
contained in |E ⊗ E/W |, where |W | is spanned by hyperplanes containing
PB(|L|). Under the involution σ V the space (E ⊗ E)∨ ∼ = E ∨ ⊗V E decomposes
2 ∨ 2 ∨ 2 ∨
into the direct product S E ⊕ E . The linear
V2 system | E | defines
the Plücker embedding of S = PB(|L|)/(σ) in | E| with the image equal
to the Reye congruence. The linear system |S 2 E | restricts on PB(|L|) to a
5-dimensional linear system |W ⊥ | ⊂ |S 2 E| = |(S 2 E ∨ )∨ |. It embeds S into
another P5 with the image not lying on a quadrics. The image is contained
in the determinantal quartic hypersurface D(W ⊥ ) of the five-dimensional
linear system of quadrics |W ⊥ |. It is equal to the singular locus of D(W ⊥ )
parameterizing quadrics of corank 2. It is a surface of degree 10 given by
partial derivatives of the equation defining D(W ⊥ ).
Remark 5.4. We constructed three different nonsingular birational models

of the Cayley quartic symmetroid D(|L|). They are D̃(|L|), St(|L), PB(|L|).
They are isomorphic to the K3-cover of an Enriques surface S which is
isomorphic to a Reye congruence of lines of bi-degree (7,3) in G1 (P3 ). We
know that the double cover of P3 branched along D(|L|) is a unirational
but non-rational 3-fold. A natural question: is the same true for the double
cover of P3 branched along the nonsingular quadric St(|L|)?
Finally, we have to prove Cayley’s theorem characterizing quartic sym-

metroids among quartics with 10 nodes. Let ηH and ηS be the divisor
classes in D̃(|L|) equal to the pre-image of a hyperplane under the mor-
˜ : D̃(|L|) → St(|L|) ⊂ |E|. One can
phisms π : D̃(|L|) → D(|L|) ⊂ |L| and st
show that
2ηS = 3ηH − r1 − · · · − r10 , (5.0.1)
where ri are the classes of the exceptional curves of the projection π. The
relation (??) characterize a minimal resolution of a quartic symmetroid from
a minimal resolution of any 10-nodal quartics (see [Dolgachev, Classical
AG]).1 The Steinerian map is given by a column of minors of the adjugate
matrix defining D(|L|). They coincide with polar cubics of D(|L|) and define
a linear subsystem of |3ηH −r1 −· · ·−r10 |. Let Qi be the quadric of corank 2
corresponding to a singular point pi of D(|L|). It is the union of two planes,
and the pre-image of Qi ∩ St(|L|) under the Steinerian map is the union of
two cubics, the residual cubics of the intersection of each of the planes with
St(|L|). These cubics are projected from pi to the union of two components
of the branch curve of the double cover.
1
For experts: the Picard lattice of the former is U ⊥ E8 (2) ⊥ h−4i and the Picard
lattice of the latter is h4i ⊥ h−2i⊕10 .
Index
1
2 -discriminant, 20 exceptional divisor, 12
k-planes, 10
Fano surface, 67
adjoint representation, 11 first polar, 13
adjoint variety, 11 flex lines, 35
adjunction formula, 31
geometric basis, 48
Brauer group, 59 Grassmann variety, 6
Clifford algebra, 8 Heisenberg group, 32

congruence of lines, 70 Hesse configuration, 35
bidegree, 71 Hesse equation, 35
class, 71 hyperelliptic curve, 49
order, 71
conic, 16 idempotent, 8
conic bundle, 26, 57 inflection points, 35
discriminant, 57 isotropic subspace
contact Fano variety, 11 maximal, 5
isotropic vector, 5
cusp, 28
Jacobian variety, 50
del Pezzo surface, 24
join, 11
anti-canonical model, 24
degree, 24 K3 surface, 30
of degree 4, 46 Klein quadrics, 19
discriminant hypersurface, 16, 39 Kummer configuration, 56, 58
Kummer surface, 55, 66
elementary divisors, 24
Kummer variety, 55
elliptic curve, 31
elliptic fibration, 28 linear system of quadrics
elliptic surface regular, 41
relatively minimal, 28 linearization, 33
elliptic surfaces
Kodaira’s types of fibers, 28 modular curve, 28
81
82 INDEX
moduli space of elliptic curves, 27 singular vector

Montesano cubic complex, 71 in characteristic 2, 19
Mordell-Weil group, 28 spinor group, 10
spinor variety, 10
orthogonal Grassmannian, 10 stable-rationality, 66
orthogonal group, 5 Steiner map, 72
osculating hyperplane, 31 stereographic projection, 14
strange point, 21
pencil of conics, 23
Plücker coordinates, 6 trisecant line, 53
Plücker map, 6 tropes, 58
polar bilinear form, 3
polarity, 19 universal Kummer family, 31
projection map, 12
its image, 13 Veronese map, 25
proper inverse transform, 12 Veronese surface, 16, 25
Prym variety, 67 Weierstrass equation, 27
quadratic cone, 17
quadratic form, 3
corank, 4
defect, 4
degenerate, 4
rank, 4
singular vector, 4
quadratic line complex, 54
quadric, 9
generator, 9
quadric surface, 16
radical, 4
resolution of singularities, 40
Reye congruence, 71
Reye line, 70
ruling, 6
Schrödinger representation, 32
Schubert calculus, 70
Segre cubic hypersurface, 66
Segre map, 16
Segre symbol, 43

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LINEAR SYSTEMS OF QUADRICS. MÜNICH.

May 24, 2020

1.1 Quadratic forms

where t1 , . . . , tn+1 is the dual basis in E ∨ (it serves as coordinates in E)

(a) q(λv) = λ2 q(v), for any λ ∈ k and v ∈ E;

(b) the function bq : E × E → k : (v, w) 7→ q(v + w) − q(v) − q(w), is

L⊥ := {v ∈ E : bq (v, w) = 0, for any w ∈ L}. (1.1.2)

This is obviously a linear subspace of E. For example, rad(bq ) = E ⊥ and

Let q be a totally degenerate P quadratic form, i.e. rad(bq ) = 0. If p 6= 2,

defined by this basis the expression of q is given by

A vector v ∈ E is called an isotropic vector of a quadratic form q if

subspace with U ∩ W = {0} and pU , pV be the projection maps from W

f1 , . . . , fr , e1 + f2 , e2 − f1 , . . . , er−1 + fr , er − fr−1 . (1.1.5)

It is immediately checked that W3 a totally isotropic and is complementary

arbitrary. This defines an open subset of the Grassmann variety isomorphic

Proposition 1.1. Let q be an non-degenerate quadratic form.

(i) If n is even, then there is only one ruling.

(ii) If n is odd, there are two rulings.

(iii) Each ruling is an irreducible closed subvariety of the Grassmann va-

Proof. (i) Let W1 and W2 be two maximal totally isotropic subspace. We

We can derive the formula for the dimension of a family of maximal

This shows that the functor K 7→ O(EK , q) is representable by an affine

Remark 1.2. Recall that a non-degenerate quadratic form q on E defines

Proposition 1.3. A quadric Q = V (q) is smooth if and only if rad(bq ) =

Proof. Fix projective coordinates so that q is given by expression (??). Re-

The point x is nonsingular or smooth if and only if dim Tx X = dimx X. The

Note that, if v0 6∈ rad(bq ), then the restriction of the quadric to this

A geometric interpretation of a totally isotropic subspace of q of dimen-

We can compute the dimension of the variety of generators in another

Ik,n = {(x, Λ) ∈ Q × Fk (Q) : x ∈ Λ}.

It comes with two projections p1 : Ik,n → Q and p2 : Ik,n → Fk (Q). The

to Fk−1,n−2 . This gives an inductive formula

dim Fk,n = dim Ik,n −k = dim Q+dim Fk−1,n−2 −k = n−1−k+dim Fk−1,n−2 .

It also shows that whenever Fk−1,n−2 is irreducible, then Fk,n is irreducible.

1.3 Birational geometry of quadrics

For any point x1 ∈ Sing(Q) with coordinates [a1 , . . . , ar , 0, . . . , 0] and a point

points in Q0 . In particular, when dim Sing(Q) = 0, the quadric Q is a cone

Γ|φ| := {([v1 ], [v2 ]) ∈ |E1 | × |E2 | : φ(v1 ) ∧ v2 = 0}.

p1 : Bl|K| |E| → |E 0 | is a projective bundle with each fiber isomorphic to

where x0 = [a1 , . . . , an+1 ].

t2n+1 + l(t1 , . . . , tn )tn+1 + q(t1 , . . . , tn ) = 0,

[t0 , t1 , t2 ] 7→ [y0 , y1 , y2 , y3 ] = [t20 , t0 t1 , t0 t2 , t21 + t2 ].

The image is a smooth quadric given by equation

Note that the pre-images of hyperplanes in P3 are conics with equations

(iii) H 2i+1 (Q, Zl ) = 0;

(iv) H 2k (Q, Zl ) is freely generated by ηQ

Proof. We use induction by n. Let X → Q be the blow-up of a point x0 ∈ Q.

where is the exceptional divisor and g : E → Z is the restriction of f to

where ci = ci (NZ/Y ) are the Chern classes of NZ/Y and r = codim(Z, Y ) =

Example 1.7. A one-dimensional quadric Q = V (q) is called a conic. It is

Example 1.8. A quadric in P3 is a quadric surface. A smooth quadric

It has two families of lines given by equations

λt1 − µt3 = µt2 + λt4 = 0,

P1 × P1 → Q, ([u0 , u1 ], [v0 , v1 ]) 7→ [u0 v0 , u0 v1 , u1 v0 , u1 v1 ]

is an isomorphism. The images of fibers of each projection P1 × P1 → P1 is

The projection from a point x0 ∈ Q to P2 extends to a regular map from

The map σ is an isomorphism outside two points y1 , y2 in P2 . The fibers over

1.4 Appendix 1:Quadrics in characteristic 2

Exercise 1.11. Let B be a skew-symmetric matrix of odd size n + 1. Let

: V k+1 (E ∨ ) → k+1 (E ∨ ). The

where i ∈ Z/nZ and n is a generator of the group µn (k) = {a ∈ k : an =

x = 0, y = 0, z = 0, Hi,j := t0 + i3 t1 + j3 t2 = 0, 0 ≤ i, j ≤ 2.