Resolution of singularities via logarithmic stacks, with a view toward the monodromy conjecture

Quek, Ming Hao

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Year:: 2023
Contributor:: Quek, Ming Hao (creator); Abramovich, Dan (Advisor); Chan, Melody (Reader); Hassett, Brendan (Reader); Brown University. Department of Mathematics (sponsor)
Genre:: theses
Subject:: Geometry, Algebraic; Arithmetical algebraic geometry; Algebraic stacks; Singularities (Mathematics); Logarithmic geometry; Birational geometry; Monodromy conjecture; Resolution of singularities; Newton diagrams
Extent:: viii, 286 p.

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Abstract:: Hironaka showed in his 1964 groundbreaking work that singularities of algebraic varieties admit a resolution in characteristic zero. Over the years, the proof of Hironaka’s theorem has seen simplifications, due to the work of Bierstone–Milman, Encinas–Villamayor, and Włodarczyk. Building on the work of Abramovich–Temkin–Włodarczyk, this thesis demonstrates that the proof becomes even simpler and much more efficient, once one allows for logarithmic algebraic stacks in the proof. In fact, an iterative resolution procedure is possible: given a singular, reduced closed subscheme of a smooth scheme over a field of characteristic zero, we resolve its singularities by iteratively blowing up the ‘‘worst singular locus’’ and immediately witnessing a visible improvement in singularities. These blow-ups will be stack-theoretic weighted blow-ups, or multi-weighted blow-ups, along locally monomial centers. We study these notions in the thesis. Finally, we show how the notion of multi-weighted blow-ups can be used to study the monodromy conjecture of Denef–Loeser, starting with the case of non-degenerate (complex) polynomials. The conjecture predicts a relationship between poles of the local motivic zeta function of a complex polynomial and ‘‘monodromy eigenvalues’’ associated with that polynomial. We improve on the previous understanding of poles of the local motivic zeta function of a non-degenerate polynomial at the origin, by constructing a resolution of singularities of its vanishing locus that is ‘‘more refined’’ than the resolution of singularities induced by its Newton polyhedron. As a consequence, we provide a new, geometric proof of the conjecture for all non-degenerate polynomials in three variables.
Notes:: Thesis (Ph. D.)--Brown University, 2023

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Quek, Ming Hao, "Resolution of singularities via logarithmic stacks, with a view toward the monodromy conjecture" (2023). Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:497wg877/

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Theses and Dissertations for the Mathematics department.
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