Functorial embedded resolution via weighted blowings up

Abramovich, Dan; Temkin, Michael; Włodarczyk, Jarosław

doi:10.2140/ant.2024.18.1557

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Functorial embedded resolution via weighted blowings up

Dan Abramovich, Michael Temkin and Jarosław Włodarczyk

Vol. 18 (2024), No. 8, 1557–1587

DOI: 10.2140/ant.2024.18.1557

Abstract

We provide a simple procedure for resolving, in characteristic 0, singularities of a variety $X$ embedded in a smooth variety $Y$ by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient.

A similar result was discovered independently by McQuillan (2020).

Keywords

resolution of singularities, weighted blowings up, algebraic stacks

Mathematical Subject Classification

Primary: 14A20, 14E15, 14M25

Milestones

Received: 15 March 2023

Revised: 6 July 2023

Accepted: 3 September 2023

Published: 18 September 2024

Authors

Dan Abramovich
Department of Mathematics Brown University Providence, RI United States

Michael Temkin
Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem Israel

Jarosław Włodarczyk
Department of Mathematics Purdue University West Lafayette, IN United States

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Vol. 18, No. 8, 2024