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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
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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