Local Newton nondegenerate Weil divisors in toric varieties

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.

37 pages, 13 figures