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.