Motivic zeta function of the Hilbert schemes of points on a surface

2020/12/13 by Luigi Pagano, Pagano, Luigi
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Alkaloids: synthesis and pharmacology

paper · pdf · doi:10.48550/arxiv.2012.07026

Abstract

Let K be a discretely-valued field. Let X→ Spec K be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes Hilbⁿ(X) over the ring of integers R⊆ K. We exploit this construction in order to compute the Motivic Zeta Function of Hilbⁿ(X) in terms of Z_X. We determine the poles of Z_Hilbⁿ(X) and study its monodromy property, showing that if the monodromy conjecture holds for X then it holds for Hilbⁿ(X) too. Sit K corpus cum absoluto ualore discreto. Sit X→ Spec K leuigata superficies cum canonico fasce congruenti O_X. In hoc scripto defecta Neroniensia paradigmata Hilbⁿ(X) schematum super annulo integrorum in K corpo, R ⊂ K, constituimus. Ex hoc, Functionem Zetam Motiuicam Z_Hilbⁿ(X), dato Z_X, computamus. Suos polos statuimus et suam monodromicam proprietatem studemus, coniectura monodromica, quae super X ualet, ualere super Hilbⁿ(X) quoque demostrando.

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