= 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.
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