Normal variety

An algebraic variety whose local rings are integrally closed domains. In particular an irreducible variety that is a normal variety has normal affine coordinate rings, enabling codimension-two extension of regular functions on a normal variety.

A singular point on a normal two-dimensional algebraic variety has an integrally closed local domain which is not a regular local ring. A rational double point is an important isolated example with a crepant minimal resolution.

A rational double point is a normal surface singularity whose minimal resolution has a configuration of smooth rational $(-2)$-curves of type $A$, $D$ or $E$. Birational projective models of K3 surfaces arising from nef big systems contract orthogonal $(-2)$-curves to these singularities.

Over an algebraically closed field of characteristic different from two, a nondegenerate quadratic form of rank at least three is irreducible, its cone has only the vertex singular, and its hypersurface ring is Cohen-Macaulay. Thus $(R_1)$ and $(S_2)$ in the Serre criterion for normality hold. In characteristic two, the reduced zero variety of ${\sum }_i{t}_i^2$ is instead the hyperplane ${\sum }_i{t}_i=0$, which is normal. The unreduced double-hyperplane scheme is not normal.