Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster

TL;DR

This paper investigates surfaces related to 3D toric idealistic clusters, providing formulas for topological zeta functions, criteria for monodromy eigenvalues, and proving conjectures on monodromy and holomorphy.

Contribution

It introduces explicit formulas for zeta functions and criteria for eigenvalues, and proves the monodromy and holomorphy conjectures for these surfaces.

Findings

Derived a formula for the topological zeta function in terms of the cluster

Established a criterion for identifying monodromy eigenvalues

Proved the monodromy and holomorphy conjectures for the class of surfaces

Abstract

In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces.