Log smoothness and polystability over valuation rings

Karim Adiprasito, Xue Liu, Igor Pak, and Michael Temkin

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Submission date: 28. Jun. 2018
Bibtex
Link to arXiv: See the arXiv entry of this preprint.

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 𝒪: 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. As a corollary we deduce that any log 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.