Functorial resolution except for toroidal locus. Toroidal compactification
Jaros{\l}aw W{\l}odarczyk

TL;DR
This paper develops a canonical resolution method for algebraic varieties in characteristic zero that preserves a specified open subset with toroidal singularities, generalizing Hironaka's desingularization and enabling toroidal compactifications.
Contribution
It introduces a functorial desingularization process that preserves a toroidal open subset, extending previous methods to more general varieties and singularities.
Findings
Existence of a canonical desingularization that preserves the toroidal locus.
Generalization of Hironaka's desingularization to varieties with toroidal singularities.
Application to toroidal equisingular compactifications.
Abstract
Let be any variety in characteristic zero. Let be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of except for V. It is a morphism , which does not modify the subset and transforms into a toroidal embedding , with singularities extending those on . Moreover, the exceptional divisor has simple normal crossings on . The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus and transforms into a nonsingular variety . The proof uses, in particular, the canonical desingularization of logarithmic varieties recently proved by Abramovich -Temkin-Wlodarczyk. It also relies on the established here canonical functorial desingularization of locally toric varieties with an unmodified open toroidal subset. As an…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Numerical Analysis Techniques · Polynomial and algebraic computation
