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Normal affine threefolds have controlled nonsmooth strata

PDF p. 15: Theorem6.7: rank-three Bertini functional

Statement

Let $k$ be algebraically closed and let $X$ be a normal affine finite-type $k$ -scheme of dimension at most three. It has finitely many open-and-closed integral normal components. Let $X_{3}$ be the union of its three-dimensional components, $W = (X_{3})_{s m / k}$ , and $D = X ∖ W$ with reduced structure. Then $W$ is smooth of pure dimension three and $D$ is closed with $\dim D \leq 2$ . On $X_{3}$ its nonsmooth locus has dimension at most one. If $X$ is pure three-dimensional, $D$ is exactly its singular locus and has dimension at most one. This formulation also handles a definition of threefold meaning merely dimension three.

Facts & Assumptions

Given: The field and scheme above; the empty stratum is allowed.

Goal: Isolate a smooth three-dimensional locus and a closed avoidance locus of dimension less than the rank three.

[F1]

External normality and perfect-field smooth-locus inputs supplies finite clopen normal components, normality implying regularity in codimension one, and equality of smooth and regular loci for reduced finite-type schemes over a perfect field.

[F2]

External finite-type dimension, rational-point and etale-fiber inputs supplies finite-type dimension and codimension formulas.

Proof

1.1F1given

The scheme is Noetherian. By [F1] its irreducible components are disjoint open-and-closed integral normal schemes, and there are finitely many because $X$ is quasi-compact. Their union $X_{3}$ is therefore clopen. The other components have dimension at most two.

2.1F1F2step 1.1∎

An algebraically closed field is perfect. On each three-dimensional component, [F1] identifies smoothness over $k$ with regularity and makes the smooth locus open. The nonsmooth locus has no points of codimension zero or one, since normality gives $(R_{1})$ . By [F2] every irreducible closed subset of that nonsmooth locus has dimension at most $3 - 2 = 1$ . Hence it is closed of dimension at most one. Combining it with the other clopen components proves that $D$ is closed of dimension at most two. Removing it leaves the smooth pure-dimensional locus $W$ . In the pure-dimensional case there are no other components, giving the sharper bound.

Depends on

Used by · 2

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