Let \(X\) be a normal projective variety over \(\CC\). Show that every irreducible component of \[ \operatorname{Sing}(X) \] has codimension at least \(2\) in \(X\). Equivalently, \(X\) is smooth in codimension one.
Solution
1. Let \(Z\subseteq X\) be an irreducible closed subset of codimension one,and let \(\eta\) be its generic point. Then\[ \dim\mco_{X,\eta}=1. \]
The point \(\eta\) corresponds to a height-one prime in any affine neighborhood of \(\eta\). The Krull dimension of the local ring at a prime is the height of that prime. Hence \[ \dim\mco_{X,\eta} = \height(\eta) = 1. \]
2. The local ring \(\mco_{X,\eta}\) is a discrete valuation ring and henceregular.
Because \(X\) is normal, every local ring of \(X\) is a normal domain. Projective varieties are Noetherian, so \[ \mco_{X,\eta} \] is a one-dimensional Noetherian normal local domain by step 1.
By the one-dimensional normality criterion D-QJ5M9, such a ring is a discrete valuation ring. A DVR is regular.
3. The variety \(X\) is smooth at every codimension-one point.
The ground field \(\CC\) is perfect. For a variety of finite type over a perfect field, regularity of the local ring is equivalent to smoothness at the corresponding point. Step 2 therefore makes every codimension-one point smooth.
4. No irreducible component of \(\operatorname{Sing}(X)\) has codimensionone.
Suppose that \[ Z\subseteq\operatorname{Sing}(X) \] were an irreducible component of codimension one, with generic point \(\eta\). Since the singular locus is closed, it contains \(\eta\).
But step 3 says that \(X\) is smooth at every codimension-one point, including \(\eta\). Thus \[ \eta\notin\operatorname{Sing}(X), \] a contradiction.
5. Therefore every irreducible component \(Z\) of the singular locussatisfies\[ \boxed{\codim(Z,X)\geq2.} \]
The singular locus is a proper closed subset of the projective variety \(X\), so every irreducible component has positive codimension. Step 4 excludes codimension one. Therefore every component has codimension at least two.
6. Q.E.D.
Steps <1>1–<1>3 establish regularity and smoothness at every codimension-one point, and steps <1>4–<1>5 convert this into the codimension bound for the singular locus.