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Independence of the embedded desingularization from the ambient embedding

Statement

Assume AC (The Axiom of Choice).

Let $U$ be an integral affine $K$ -variety of finite type over a field $K$ of characteristic zero and let $φ_{1} ⁣ : U ↪ X_{1}$ and $φ_{2} ⁣ : U ↪ X_{2}$ be two closed immersions into smooth affine $K$ -schemes (Closed immersions of schemes). Let ${\tilde{U}}_{i} \subseteq {\tilde{X}}_{i}$ be the canonical embedded desingularizations of $U$ in $X_{i}$ (Bravo-Villamayor strengthening of embedded desingularization). Then the induced desingularizations ${\tilde{U}}_{1} \to U$ and ${\tilde{U}}_{2} \to U$ are canonically isomorphic over $U$ . More precisely, the amalgamated embeddings $Ψ_{0}, Ψ_{1}, Ψ_{2} ⁣ : U \to A^{2 n}$ of any two presentations of $U$ as a closed subvariety of affine space are related by automorphisms $Φ_{1}, Φ_{2}$ of $A^{2 n}$ with $Φ_{i} Ψ_{0} = Ψ_{i}$ , and commutativity of embedded desingularization with ambient embeddings identifies the resulting resolutions over $U$ .

Facts & Assumptions

Given: An affine $K$ -variety $U$ of finite type over a field $K$ of characteristic zero and two closed embeddings $φ_{1} ⁣ : U ↪ X_{1}$ , $φ_{2} ⁣ : U ↪ X_{2}$ into smooth affine $K$ -varieties.

[A1]

The Axiom of Choice: AC is assumed for the canonical-resolution consumer clauses and their cited AC-dependent construction suppliers.

[F1]

Weak embedded desingularization in characteristic zero, Bravo-Villamayor strengthening of embedded desingularization: embedded desingularization of $U ↪ X$ is compatible with closed embeddings of smooth ambient varieties: if $X ↪ X^{'}$ is a closed embedding of smooth varieties and $U^{'}$ denotes the closure extension, then the desingularization of $U$ in $X^{'}$ restricts to the desingularization in $X$ .

[F2]

Weak embedded desingularization in characteristic zero, Bravo-Villamayor strengthening of embedded desingularization: for each closed embedding $U ↪ X$ there is a canonical embedded desingularization $\tilde{U} \subset \tilde{X}$ of $U$ in $X$ .

[F3]

Closed immersions of schemes, Locally finite type and finite type morphisms: the embeddings $φ_{i}$ may be composed with closed embeddings $X_{i} ↪ A^{n}$ (for $n$ large enough to accommodate finite generating lists for both affine coordinate rings) to obtain embeddings $ψ_{i} φ_{i} ⁣ : U \to A^{n}$ ; the coordinate functions of $X_{i}$ express the two families of generators of $K [U]$ .

[F4]

The automorphism lemma (source Lemma 4.8.1). If $g_{1}, \dots, g_{n}, h_{1}, \dots, h_{n}$ generate $K [U]$ and $Ψ_{0} (x) = (g, h)$ , $Ψ_{1} (x) = (g, 0)$ , $Ψ_{2} (x) = (0, h)$ are the three embeddings $U \to A^{2 n}$ , then choosing polynomials $w_{i} (h) = g_{i}$ and $v_{i} (g) = h_{i}$ (possible because the $h$ 's generate $K [U]$ ) gives automorphisms $Φ_{1} (x, y) = (x, y - v (x))$ and $Φ_{2} (x, y) = (x - w (y), y)$ of $A^{2 n}$ with $Φ_{i} Ψ_{0} = Ψ_{i}$ ; both are polynomial automorphisms with polynomial inverse.

Proof

1.1A1F3F4

Reduction to a common ambient space. By [F3], compose the given embeddings with embeddings of $X_{i}$ into a common $A^{n}$ . Their coordinate functions give two generating lists $g, h$ for $K [U]$ . In $A^{2 n}$ put $Ψ_{0} = (g, h)$ , $Ψ_{1} = (g, 0)$ and $Ψ_{2} = (0, h)$ . By [F4], $Φ_{i} Ψ_{0} = Ψ_{i}$ , hence $Φ_{i}^{- 1} Ψ_{i} = Ψ_{0}$ . Thus after the coordinate inclusions $A^{n} ↪ A^{2 n}$ , the inverse polynomial automorphisms carry each presentation to the same closed embedding $Ψ_{0}$ .

2.1A1F1F2F4step 1.1∎

Comparison of the resolutions. Use [F2] to resolve the common embedding $Ψ_{0}$ . The compatibility with closed smooth ambient embeddings in [F1] identifies the resolution induced by each $X_{i}$ with that induced by its coordinate inclusion in $A^{2 n}$ . Transport along $Φ_{i}^{- 1}$ , using the naturality of the canonical construction under ambient automorphisms, then identifies it with the resolution of $Ψ_{0}$ . Both comparisons are over $U$ , so their composite canonically identifies ${\tilde{U}}_{1} \to U$ with ${\tilde{U}}_{2} \to U$ .

Depends on

Used by

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jaroslaw Wlodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779-822; author's arXiv version math/0401401 (28 pp., dated October 25, 2018) (standard reference, not scraped)