# normal irreducible varieties are nonsingular in codimension 1

###### Theorem 1.

Let $X$ be a normal irreducible variety. The singular set $S\subset X$ has codimension 2 or more.

###### Proof.

Assume not. We may assume $X$ is affine, since codimension is local. Now let $\mathfrak{u}$ be the ideal of functions vanishing on $S$. This is an ideal of height 1, so the local ring of $Y$, $\mathcal{O}_{S}=A(X)_{\mathfrak{u}}$, where $A(X)$ is the affine ring of $X$, is a 1-dimensional local ring, and integrally closed, since $X$ is normal. Any integrally closed 1-dimensional local domain is a DVR, and thus regular. But $S$ is the singular set, so its local ring is not regular, a contradiction. ∎

Title normal irreducible varieties are nonsingular in codimension 1 NormalIrreducibleVarietiesAreNonsingularInCodimension1 2013-03-22 13:20:20 2013-03-22 13:20:20 archibal (4430) archibal (4430) 7 archibal (4430) Theorem msc 14A99