Motivic invariants in birational geometry and singularity theory
File(s)
Author(s)
Moe, Simen
Type
Thesis
Abstract
This thesis is mainly concerned with computing motivic invariants in two particular cases. Given two polynomials f and g in disjoint sets of variables, we compute the motivic zeta function of f + g in terms of strictly toroidal resolutions of f and g. Using this, we show that the motivic Thom-Sebastiani formula of Denef–Loeser holds already in the Grothendieck ring of varieties without inverting the affine line. We also compute weak strictly toroidal resolutions for singularities of the form P(f,g) for certain non-degenerate polynomials P, and show that the motivic monodromy conjecture holds in this case.
Limits of these zeta functions have interesting properties as well, and Nicaise– Shinder showed that they can be used as an obstruction to stable rationality. Building on the combinatorial framework of Nicaise–Ottem to study stable rationality of hypersurfaces in toric varieties (in terms of the associated polytope), we give a purely combinatorial strategy to prove stable irrationality of hypersurfaces in toric varieties. We apply this strategy to give many new examples of stably irrational hypersurfaces in projective space and products of projective spaces.
Limits of these zeta functions have interesting properties as well, and Nicaise– Shinder showed that they can be used as an obstruction to stable rationality. Building on the combinatorial framework of Nicaise–Ottem to study stable rationality of hypersurfaces in toric varieties (in terms of the associated polytope), we give a purely combinatorial strategy to prove stable irrationality of hypersurfaces in toric varieties. We apply this strategy to give many new examples of stably irrational hypersurfaces in projective space and products of projective spaces.
Version
Open Access
Date Issued
2024-06-25
Date Awarded
01/01/2025
License URL
Advisor
Nicaise, Johannes
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)