In the mathematical field of algebraic geometry, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. 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.
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. 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. 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.
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