Local Newton nondegenerate Weil divisors in toric varieties

TL;DR

This paper develops a combinatorial and geometric theory for Newton nondegenerate local Weil divisors in toric varieties, characterizing their properties and invariants through Newton diagrams and toric combinatorics.

Contribution

It introduces a new framework linking Newton diagrams with the properties of Weil divisors in toric varieties, including formulas for invariants and resolution structures.

Findings

Characterization of normality, Gorenstein property, and Cartier divisors via Newton diagrams.

Formulas for delta-invariant and cohomology groups of resolutions.

Resolution graphs and genus formulas for 2-dimensional cases.

Abstract

We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution $\tilde{X}\to X $ and the corresponding canonical divisor. We provide combinatorial formulae for the delta--invariant $\delta(X,0)$ and for the cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case $\dim(X,0)=2$, we provide the (canonical) resolution graph from the Newton diagram and we also prove that if such a Weil divisor is normal and Gorenstein, and the link is a rational homology sphere, then the geometric genus is given by the minimal path cohomology, a topological invariant of the link.