Quadric (algebraic geometry)

In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface

${\displaystyle xy=zw}$

in projective space ${\displaystyle {\mathbf {P} }^{3}}$ over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry.

Many properties of quadrics hold more generally for projective homogeneous varieties. Another generalization of quadrics is provided by Fano varieties.

By definition, a quadric X of dimension n over a field k is the subspace of ${\displaystyle \mathbf {P} ^{n+1}}$ defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables ${\displaystyle x_{0},\ldots ,x_{n+1}}$. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.) If q is the product of two linear forms, then X is the union of two hyperplanes. It is common to assume that ${\displaystyle n\geq 1}$ and q is irreducible, which excludes that special case.

Here algebraic varieties over a field k are considered as a special class of schemes over k. When k is algebraically closed, one can also think of a projective variety in a more elementary way, as a subset of ${\displaystyle {\mathbf {P} }^{N}(k)=(k^{N+1}-0)/k^{*}}$ defined by homogeneous polynomial equations with coefficients in k.

If q can be written (after some linear change of coordinates) as a polynomial in a proper subset of the variables, then X is the projective cone over a lower-dimensional quadric. It is reasonable to focus attention on the case where X is not a cone. For k of characteristic not 2, X is not a cone if and only if X is smooth over k. When k has characteristic not 2, smoothness of a quadric is also equivalent to the Hessian matrix of q having nonzero determinant, or to the associated bilinear form b(x,y) = q(x+y) – q(x) – q(y) being nondegenerate. In general, for k of characteristic not 2, the rank of a quadric means the rank of the Hessian matrix. A quadric of rank r is an iterated cone over a smooth quadric of dimension r − 2.^[1]

It is a fundamental result that a smooth quadric over a field k is rational over k if and only if X has a k-rational point.^[2] That is, if there is a solution of the equation q = 0 of the form ${\displaystyle (a_{0},\ldots ,a_{n+1})}$ with ${\displaystyle a_{0},\ldots ,a_{n+1}}$ in k, not all zero (hence corresponding to a point in projective space), then there is a one-to-one correspondence defined by rational functions over k between ${\displaystyle {\mathbf {P} }^{n}}$ minus a lower-dimensional subset and X minus a lower-dimensional subset. For example, if k is infinite, it follows that if X has one k-rational point then it has infinitely many. This equivalence is proved by stereographic projection. In particular, every quadric over an algebraically closed field is rational.

A quadric over a field k is called isotropic if it has a k-rational point. An example of an anisotropic quadric is the quadric

${\displaystyle x_{0}^{2}+x_{1}^{2}+\cdots +x_{n+1}^{2}=0}$

in projective space ${\displaystyle {\mathbf {P} }^{n+1}}$ over the real numbers R.

A central part of the geometry of quadrics is the study of the linear spaces that they contain. (In the context of projective geometry, a linear subspace of ${\displaystyle {\mathbf {P} }^{N}}$ is isomorphic to ${\displaystyle {\mathbf {P} }^{a}}$ for some ${\displaystyle a\leq N}$.) A key point is that every linear space contained in a smooth quadric has dimension at most half the dimension of the quadric. Moreover, when k is algebraically closed, this is an optimal bound, meaning that every smooth quadric of dimension n over k contains a linear subspace of dimension ${\displaystyle \lfloor n/2\rfloor }$.^[3]

Over any field k, a smooth quadric of dimension n is called split if it contains a linear space of dimension ${\displaystyle \lfloor n/2\rfloor }$ over k. Thus every smooth quadric over an algebraically closed field is split. If a quadric X over a field k is split, then it can be written (after a linear change of coordinates) as

${\displaystyle x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m-2}x_{2m-1}+x_{2m}^{2}=0}$

if X has dimension 2m − 1, or

${\displaystyle x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m}x_{2m+1}=0}$

if X has dimension 2m.^[4] In particular, over an algebraically closed field, there is only one smooth quadric of each dimension, up to isomorphism.

For many applications, it is important to describe the space Y of all linear subspaces of maximal dimension in a given smooth quadric X. (For clarity, assume that X is split over k.) A striking phenomenon is that Y is connected if X has odd dimension, whereas it has two connected components if X has even dimension. That is, there are two different "types" of maximal linear spaces in X when X has even dimension. The two families can be described by: for a smooth quadric X of dimension 2m, fix one m-plane Q contained in X. Then the two types of m-planes P contained in X are distinguished by whether the dimension of the intersection ${\displaystyle P\cap Q}$ is even or odd.^[5] (The dimension of the empty set is taken to be −1 here.)

Let X be a split quadric over a field k. (In particular, X can be any smooth quadric over an algebraically closed field.) In low dimensions, X and the linear spaces it contains can be described as follows.

• A quadric curve in ${\displaystyle \mathbf {P} ^{2}}$ is called a conic. A split conic over k is isomorphic to the projective line ${\displaystyle \mathbf {P} ^{1}}$ over k, embedded in ${\displaystyle \mathbf {P} ^{2}}$ by the 2nd Veronese embedding.^[6] (For example, ellipses, parabolas and hyperbolas are different kinds of conics in the affine plane over R, but their closures in the projective plane are all isomorphic to ${\displaystyle \mathbf {P} ^{1}}$ over R.)

• A split quadric surface X is isomorphic to ${\displaystyle \mathbf {P} ^{1}\times \mathbf {P} ^{1}}$, embedded in ${\displaystyle \mathbf {P} ^{3}}$ by the Segre embedding. The space of lines in the quadric surface X has two connected components, each isomorphic to ${\displaystyle \mathbf {P} ^{1}}$.^[7]

• A split quadric 3-fold X can be viewed as an isotropic Grassmannian for the symplectic group Sp(4,k). (This is related to the exceptional isomorphism of linear algebraic groups between SO(5,k) and ${\displaystyle \operatorname {Sp} (4,k)/\{\pm 1\}}$.) Namely, given a 4-dimensional vector space V with a symplectic form, the quadric 3-fold X can be identified with the space LGr(2,4) of 2-planes in V on which the form restricts to zero. Furthermore, the space of lines in the quadric 3-fold X is isomorphic to ${\displaystyle \mathbf {P} ^{3}}$.^[8]

• A split quadric 4-fold X can be viewed as the Grassmannian Gr(2,4), the space of 2-planes in a 4-dimensional vector space (or equivalently, of lines in ${\displaystyle \mathbf {P} ^{3}}$). (This is related to the exceptional isomorphism of linear algebraic groups between SO(6,k) and ${\displaystyle \operatorname {SL} (4,k)/\{\pm 1\}}$.) The space of 2-planes in the quadric 4-fold X has two connected components, each isomorphic to ${\displaystyle \mathbf {P} ^{3}}$.^[9]

• The space of 2-planes in a split quadric 5-fold is isomorphic to a split quadric 6-fold. Likewise, both components of the space of 3-planes in a split quadric 6-fold are isomorphic to a split quadric 6-fold. (This is related to the phenomenon of triality for the group Spin(8).)

As these examples suggest, the space of m-planes in a split quadric of dimension 2m always has two connected components, each isomorphic to the isotropic Grassmannian of (m − 1)-planes in a split quadric of dimension 2m − 1.^[10] Any reflection in the orthogonal group maps one component isomorphically to the other.

A smooth quadric over a field k is a projective homogeneous variety for the orthogonal group (and for the special orthogonal group), viewed as linear algebraic groups over k. Like any projective homogeneous variety for a split reductive group, a split quadric X has an algebraic cell decomposition, known as the Bruhat decomposition. (In particular, this applies to every smooth quadric over an algebraically closed field.) That is, X can be written as a finite union of disjoint subsets that are isomorphic to affine spaces over k of various dimensions. (For projective homogeneous varieties, the cells are called Schubert cells, and their closures are called Schubert varieties.) Cellular varieties are very special among all algebraic varieties. For example, a cellular variety is rational, and (for k = C) the Hodge theory of a smooth projective cellular variety is trivial, in the sense that ${\displaystyle h^{p,q}(X)=0}$ for ${\displaystyle p\neq q}$. For a cellular variety, the Chow group of algebraic cycles on X is the free abelian group on the set of cells, as is the integral homology of X (if k = C).^[11]

A split quadric X of dimension n has only one cell of each dimension r, except in the middle dimension of an even-dimensional quadric, where there are two cells. The corresponding cell closures (Schubert varieties) are:^[12]

• For ${\displaystyle 0\leq r<n/2}$, a linear space ${\displaystyle \mathbf {P} ^{r}}$ contained in X.

• For r = n/2, both Schubert varieties are linear spaces ${\displaystyle \mathbf {P} ^{r}}$ contained in X, one from each of the two families of middle-dimensional linear spaces (as described above).

• For ${\displaystyle n/2<r\leq n}$, the Schubert variety of dimension r is the intersection of X with a linear space of dimension r + 1 in ${\displaystyle \mathbf {P} ^{n+1}}$; so it is an r-dimensional quadric. It is the iterated cone over a smooth quadric of dimension 2r − n.

Using the Bruhat decomposition, it is straightforward to compute the Chow ring of a split quadric of dimension n over a field, as follows.^[13] When the base field is the complex numbers, this is also the integral cohomology ring of a smooth quadric, with ${\displaystyle CH^{j}}$ mapping isomorphically to ${\displaystyle H^{2j}}$. (The cohomology in odd degrees is zero.)

• For n = 2m − 1, ${\displaystyle CH^{*}(X)\cong \mathbb {Z} [h,l]/(h^{m}-2l,l^{2})}$, where |h| = 1 and |l| = m.

• For n = 2m, ${\displaystyle CH^{*}(X)\cong \mathbb {Z} [h,l]/(h^{m+1}-2hl,l^{2}-ah^{m}l)}$, where |h| = 1 and |l| = m, and a is 0 for m odd and 1 for m even.

Here h is the class of a hyperplane section and l is the class of a maximal linear subspace of X. (For n = 2m, the class of the other type of maximal linear subspace is ${\displaystyle h^{m}-l}$.) This calculation shows the importance of the linear subspaces of a quadric: the Chow ring of all algebraic cycles on X is generated by the "obvious" element h (pulled back from the class ${\displaystyle c_{1}O(1)}$ of a hyperplane in ${\displaystyle {\mathbf {P} }^{n+1}}$) together with the class of a maximal linear subspace of X.

The space of r-planes in a smooth n-dimensional quadric (like the quadric itself) is a projective homogeneous variety, known as the isotropic Grassmannian or orthogonal Grassmannian OGr(r + 1, n + 2). (The numbering refers to the dimensions of the corresponding vector spaces. In the case of middle-dimensional linear subspaces of a quadric of even dimension 2m, one writes ${\displaystyle \operatorname {OGr} _{+}(m+1,2m+2)}$ for one of the two connected components.) As a result, the isotropic Grassmannians of a split quadric over a field also have algebraic cell decompositions.

The isotropic Grassmannian W = OGr(m,2m + 1) of (m − 1)-planes in a smooth quadric of dimension 2m − 1 may also be viewed as the variety of Projective pure spinors, or simple spinor variety,^[14][15] of dimension m(m + 1)/2. (Another description of the pure spinor variety is as ${\displaystyle \operatorname {OGr} _{+}(m+1,2m+2)}$.^[10]) To explain the name: the smallest SO(2m + 1)-equivariant projective embedding of W lands in projective space of dimension ${\displaystyle 2^{m}-1}$.^[16] The action of SO(2m + 1) on this projective space does not come from a linear representation of SO(2m+1) over k, but rather from a representation of its simply connected double cover, the spin group Spin(2m + 1) over k. This is called the spin representation of Spin(2m + 1), of dimension ${\displaystyle 2^{m}}$.

Over the complex numbers, the isotropic Grassmannian OGr(r + 1, n + 2) of r-planes in an n-dimensional quadric X is a homogeneous space for the complex algebraic group ${\displaystyle G=\operatorname {SO} (n+2,\mathbf {C} )}$, and also for its maximal compact subgroup, the compact Lie group SO(n + 2). From the latter point of view, this isotropic Grassmannian is

${\displaystyle \operatorname {SO} (n+2)/(\operatorname {U} (r+1)\times \operatorname {SO} (n-2r)),}$

where U(r+1) is the unitary group. For r = 0, the isotropic Grassmannian is the quadric itself, which can therefore be viewed as

${\displaystyle \operatorname {SO} (n+2)/(\operatorname {U} (1)\times \operatorname {SO} (n)).}$

For example, the complex projectivized pure spinor variety OGr(m, 2m + 1) can be viewed as SO(2m + 1)/U(m), and also as SO(2m+2)/U(m+1). These descriptions can be used to compute the cohomology ring (or equivalently the Chow ring) of the spinor variety:

${\displaystyle CH^{*}\operatorname {OGr} (m,2m+1)\cong \mathbb {Z} [e_{1},\ldots ,e_{m}]/(e_{j}^{2}-2e_{j-1}e_{j+1}+2e_{j-2}e_{j+2}-\cdots +(-1)^{j}e_{2j}=0{\text{ for all }}j),}$

where the Chern classes ${\displaystyle c_{j}}$ of the natural rank-m vector bundle are equal to ${\displaystyle 2e_{j}}$.^[17] Here ${\displaystyle e_{j}}$ is understood to mean 0 for j > m.

The spinor bundles play a special role among all vector bundles on a quadric, analogous to the maximal linear subspaces among all subvarieties of a quadric. To describe these bundles, let X be a split quadric of dimension n over a field k. The special orthogonal group SO(n+2) over k acts on X, and therefore so does its double cover, the spin group G = Spin(n+2) over k. In these terms, X is a homogeneous space G/P, where P is a maximal parabolic subgroup of G. The semisimple part of P is the spin group Spin(n), and there is a standard way to extend the spin representations of Spin(n) to representations of P. (There are two spin representations ${\displaystyle V_{+},V_{-}}$ for n = 2m, each of dimension ${\displaystyle 2^{m-1}}$, and one spin representation V for n = 2m − 1, of dimension ${\displaystyle 2^{m-1}}$.) Then the spinor bundles on the quadric X = G/P are defined as the G-equivariant vector bundles associated to these representations of P. So there are two spinor bundles ${\displaystyle S_{+},S_{-}}$ of rank ${\displaystyle 2^{m-1}}$ for n = 2m, and one spinor bundle S of rank ${\displaystyle 2^{m-1}}$ for n = 2m − 1. For n even, any reflection in the orthogonal group switches the two spinor bundles on X.^[16]

For example, the two spinor bundles on a quadric surface ${\displaystyle X\cong \mathbf {P} ^{1}\times \mathbf {P} ^{1}}$ are the line bundles O(−1,0) and O(0,−1). The spinor bundle on a quadric 3-fold X is the natural rank-2 subbundle on X viewed as the isotropic Grassmannian of 2-planes in a 4-dimensional symplectic vector space.

To indicate the significance of the spinor bundles: Mikhail Kapranov showed that the bounded derived category of coherent sheaves on a split quadric X over a field k has a full exceptional collection involving the spinor bundles, along with the "obvious" line bundles O(j) restricted from projective space:

${\displaystyle D^{b}(X)=\langle S_{+},S_{-},O,O(1),\ldots ,O(n-1)\rangle }$

if n is even, and

${\displaystyle D^{b}(X)=\langle S,O,O(1),\ldots ,O(n-1)\rangle }$

if n is odd.^[18] Concretely, this implies the split case of Richard Swan's calculation of the Grothendieck group of algebraic vector bundles on a smooth quadric; it is the free abelian group

${\displaystyle K_{0}(X)=\mathbb {Z} \{S_{+},S_{-},O,O(1),\ldots ,O(n-1)\}}$

for n even, and

${\displaystyle K_{0}(X)=\mathbb {Z} \{S,O,O(1),\ldots ,O(n-1)\}}$

for n odd.^[19] When k = C, the topological K-group ${\displaystyle K^{0}(X)}$ (of continuous complex vector bundles on the quadric X) is given by the same formula, and ${\displaystyle K^{1}(X)}$ is zero.

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