 Normality of a quadratic cone

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.