Quadric (algebraic geometry)

Many properties of quadrics hold more generally for projective homogeneous varieties.

defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables

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.

[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.

A quadric over a field k is called isotropic if it has a k-rational point.

over the real numbers R. A central part of the geometry of quadrics is the study of the linear spaces that they contain.

over k. Thus every smooth quadric over an algebraically closed field is split.

[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.

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

In low dimensions, X and the linear spaces it contains can be described as follows.

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.

(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.

The corresponding cell closures (Schubert varieties) are:[12] 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

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

(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

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.

From the latter point of view, this isotropic Grassmannian is where U(r+1) is the unitary group.

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.

For n even, any reflection in the orthogonal group switches the two spinor bundles on X.

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: if n is even, and 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 for n even, and for n odd.

(of continuous complex vector bundles on the quadric X) is given by the same formula, and