Extending Triangulations and Semistable Reduction

Abramovich, Dan; Rojas, J. Maurice

doi:10.1142/9789812778031_0001

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2018

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“…In fact, our main combinatorial result, Theorem 4.5.2, can be viewed as a local on the base version of that conjecture, and the two problems are tightly related. In particular, Dan Abramovich pointed out that the first stage of our construction, existence of a polysimplicial subdivision, was established in the global setting in [AR02]. The conjecture of Abramovich-Karu can be treated by our method as well, and is the subject of [ALT18].…”

Section: Resultsmentioning

confidence: 99%

Log smoothness and polystability over valuation rings

Adiprasito,

Liu,

Pak

et al. 2018

Preprint

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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 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 p n . The core of our proof is a subdivision result for polyhedral complexes satisfying certain rationality conditions.

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Section: Resultsmentioning

confidence: 99%