TITLE: Checking whether a variety is normal
QUESTION [13 upvotes]: I am looking to check whether the hypersurface in $A^{n}$ defined by $x_{1}^{2} + x_2^{2} + .... + x_n^{2} = 0$ is a normal variety.....In general, are there any nice sufficiency conditions to prove normality?

REPLY [3 votes]: Given a Cohen-Macaulay affine ring $R$ over a perfect field, it satisfies $S_k$ for all $k$. Since $R$ is equidimensional, Jacobian criterion says that the singular locus is cut out by the Jacobian ideal $J$. It is then sufficient and necessary to have codim $J \ge 2$ in $R$ for $R$ to satisfy $R_1$. 
Assuming characteristic is not 2, the coordinate ring $S = k[x_1,\cdots,x_n]/(x_1^2+\cdots+ x_r^2)$ is cut out by a non-zerodivisor and is thus Cohen-Macaulay. Its singular locus is cut out by $I  = (x_1,\cdots, x_r)$ which has codim $r-1$ in $S$. Therefore $S$ is normal iff $r\ge 3$.