Log smoothness and polystability over valuation rings

2018/06/24 by Karim Adiprasito, Adiprasito, Karim, Gaku Liu +5
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG

paper · pdf · doi:10.48550/arxiv.1806.09168

40 pages, revisions based on comments from referee

openalex publication_date 2018/06/24 · arxiv created 2019/04/30 · arxiv updated 2019/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let O be a valuation ring of height one of residual characteristic exponent p and with algebraically closed field of fractions. Our main result provides a best possible resolution of the monoidal structure M_X of a log variety X over \calO with a vertical log structure: there exists a log modification Y→ X such that the monoidal structure of Y is polystable. In particular, if X is log smooth over O, then Y is polystable with a smooth generic fiber. As a corollary we deduce that any variety over O possesses a polystable alteration of degreee pⁿ. The core of our proof is a subdivision result for polyhedral complexes satisfying certain rationality conditions.

Citations

• Foundations for almost ring theory -- Release 7.5

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