Quadric (algebraic geometry)

Subspace defined by a polynomial of degree 2 over a field

This article is about quadrics in algebraic geometry. For quadrics over the real numbers, see quadric.

The two families of lines on a smooth (split) quadric surface

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

x y = z w {\displaystyle xy=zw}

in projective space  P 3 {\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  P n + 1 {\displaystyle \mathbf {P} ^{n+1}} defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables  x 0 , … , x n + 1 {\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  n ≥ 1 {\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  P N ( k ) = ( k N + 1 − 0 ) / k ∗ {\displaystyle {\mathbf {P} }^{N}(k)=(k^{N+1}-0)/k^{*}} defined by homogeneous polynomial equations with coefficients in k.

A singular quadric surface, the cone over a smooth conic curve

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  ( a 0 , … , a n + 1 ) {\displaystyle (a_{0},\ldots ,a_{n+1})} with  a 0 , … , a n + 1 {\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  P n {\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

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

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

## Linear subspaces of quadrics

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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  P N {\displaystyle {\mathbf {P} }^{N}} is isomorphic to  P a {\displaystyle {\mathbf {P} }^{a}} for some  a ≤ N {\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  ⌊ n / 2 ⌋ {\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  ⌊ n / 2 ⌋ {\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

x 0 x 1 + x 2 x 3 + ⋯ + x 2 m − 2 x 2 m − 1 + x 2 m 2 = 0 {\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

x 0 x 1 + x 2 x 3 + ⋯ + x 2 m x 2 m + 1 = 0 {\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 P ∩ Q {\displaystyle P\cap Q} is even or odd.[5] (The dimension of the empty set is taken to be −1 here.)

## Low-dimensional quadrics

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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  P 2 {\displaystyle \mathbf {P} ^{2}} is called a conic. A split conic over k is isomorphic to the projective line  P 1 {\displaystyle \mathbf {P} ^{1}} over k, embedded in  P 2 {\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  P 1 {\displaystyle \mathbf {P} ^{1}} over R.)

* A split quadric surface X is isomorphic to  P 1 × P 1 {\displaystyle \mathbf {P} ^{1}\times \mathbf {P} ^{1}} , embedded in  P 3 {\displaystyle \mathbf {P} ^{3}} by the Segre embedding. The space of lines in the quadric surface X has two connected components, each isomorphic to  P 1 {\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  Sp ⁡ ( 4 , k ) / { ± 1 } {\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  P 3 {\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  P 3 {\displaystyle \mathbf {P} ^{3}} ). (This is related to the exceptional isomorphism of linear algebraic groups between SO(6,k) and  SL ⁡ ( 4 , k ) / { ± 1 } {\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  P 3 {\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.

## The Bruhat decomposition

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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  h p , q ( X ) = 0 {\displaystyle h^{p,q}(X)=0} for  p ≠ q {\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  0 ≤ r < n / 2 {\displaystyle 0\leq r<n/2} , a linear space  P r {\displaystyle \mathbf {P} ^{r}} contained in X.

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

* For  n / 2 < r ≤ n {\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  P n + 1 {\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  C H j {\displaystyle CH^{j}} mapping isomorphically to  H 2 j {\displaystyle H^{2j}} . (The cohomology in odd degrees is zero.)

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

* For n = 2m,  C H ∗ ( X ) ≅ Z [ h , l ] / ( h m + 1 − 2 h l , l 2 − a h m l ) {\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  h m − l {\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  c 1 O ( 1 ) {\displaystyle c_{1}O(1)} of a hyperplane in  P n + 1 {\displaystyle {\mathbf {P} }^{n+1}} ) together with the class of a maximal linear subspace of X.

## Isotropic Grassmannians and the projective pure spinor variety

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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  OGr + ⁡ ( m + 1 , 2 m + 2 ) {\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  OGr + ⁡ ( m + 1 , 2 m + 2 ) {\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  2 m − 1 {\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  2 m {\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  G = SO ⁡ ( n + 2 , C ) {\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

SO ⁡ ( n + 2 ) / ( U ⁡ ( r + 1 ) × SO ⁡ ( n − 2 r ) ) , {\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

SO ⁡ ( n + 2 ) / ( U ⁡ ( 1 ) × SO ⁡ ( n ) ) . {\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:

C H ∗ OGr ⁡ ( m , 2 m + 1 ) ≅ Z [ e 1 , … , e m ] / ( e j 2 − 2 e j − 1 e j + 1 + 2 e j − 2 e j + 2 − ⋯ + ( − 1 ) j e 2 j = 0  for all  j ) , {\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 c j {\displaystyle c_{j}} of the natural rank-m vector bundle are equal to  2 e j {\displaystyle 2e_{j}} .[17] Here  e j {\displaystyle e_{j}} is understood to mean 0 for j > m.

## Spinor bundles on quadrics

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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  V + , V − {\displaystyle V_{+},V_{-}} for n = 2m, each of dimension  2 m − 1 {\displaystyle 2^{m-1}} , and one spin representation V for n = 2m − 1, of dimension  2 m − 1 {\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  S + , S − {\displaystyle S_{+},S_{-}} of rank  2 m − 1 {\displaystyle 2^{m-1}} for n = 2m, and one spinor bundle S of rank  2 m − 1 {\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  X ≅ P 1 × P 1 {\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:

D b ( X ) = ⟨ S + , S − , O , O ( 1 ) , … , O ( n − 1 ) ⟩ {\displaystyle D^{b}(X)=\langle S_{+},S_{-},O,O(1),\ldots ,O(n-1)\rangle }

if n is even, and

D b ( X ) = ⟨ S , O , O ( 1 ) , … , O ( n − 1 ) ⟩ {\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

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

for n even, and

K 0 ( X ) = Z { S , O , O ( 1 ) , … , O ( n − 1 ) } {\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 K 0 ( X ) {\displaystyle K^{0}(X)} (of continuous complex vector bundles on the quadric X) is given by the same formula, and  K 1 ( X ) {\displaystyle K^{1}(X)} is zero.

## Notes

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1. ^ Harris (1995), Example 3.3.
2. ^ Elman, Karpenko & Merkurjev (2008), Proposition 22.9.
3. ^ Harris (1995), Theorem 22.13.
4. ^ Elman, Karpenko, & Merkurjev (2008), Proposition 7.28.
5. ^ Harris (1995), Theorem 22.14.
6. ^ Harris (1995), Lecture 22, p. 284.
7. ^ Harris (1995), Lecture 22, p. 285.
8. ^ Harris (1995), Exercise 22.6.
9. ^ Harris (1995), Example 22.7.
10. ^ a b Harris (1995), Theorem 22.14.
11. ^ Fulton (1998), Example 19.1.11.
12. ^ Elman, Karpenko & Merkurjev (2008), Proposition 68.1.
13. ^ Elman, Karpenko, & Merkurjev (2008), Exercise 68.3.
14. ^ Cartan, Élie (1981) [1938], The theory of spinors, New York: Dover Publications, ISBN 978-0-486-64070-9, MR 0631850
15. ^ Chevalley, Claude (1996) [1954]. The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 978-3-540-57063-9.
16. ^ a b Ottaviani (1988), section 1.
17. ^ Mimura & Toda (1991), Theorem III.6.11.
18. ^ Kapranov (1988), Theorem 4.10.
19. ^ Swan (1985), Theorem 1.

## References

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* Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008), Algebraic and geometric theory of quadratic forms, American Mathematical Society, ISBN 978-0-8218-4329-1, MR 2427530
* Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
* Harris, Joe (1995), Algebraic geometry: a first course, Springer-Verlag, ISBN 0-387-97716-3, MR 1416564
* Kapranov, Mikhail (1988), "On the derived categories of coherent sheaves on some homogeneous spaces", Inventiones Mathematicae, 92 (3): 479–508, Bibcode:1988InMat..92..479K, doi:10.1007/BF01393744, MR 0939472, S2CID 119584668
* Mimura, Mamoru; Toda, Hirosi (1992), Topology of Lie groups, American Mathematical Society, ISBN 978-0821813423, MR 1122592
* Ottaviani, Giorgio (1988), "Spinor bundles on quadrics", Transactions of the American Mathematical Society, 307: 301–316, doi:10.1090/S0002-9947-1988-0936818-5, MR 0936818
* Swan, Richard (1985), "K-theory of quadric hypersurfaces", Annals of Mathematics, 122 (1): 113–153, doi:10.2307/1971371, JSTOR 1971371, MR 0799254