Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

TL;DR

This paper presents a new formula for computing the Euler characteristic of Weil divisors on complete simplicial toric varieties, linking Cox rings, Stanley-Reisner rings, and Alexander duality.

Contribution

It introduces a novel approach combining Cox ring theory and Alexander duality to compute Euler characteristics via Stanley-Reisner rings.

Findings

Derived a formula connecting Euler characteristic with graded Cox ring pieces.

Utilized Alexander duality to relate irrelevant ideals to Stanley-Reisner ideals.

Extended existing work by integrating cohomology and combinatorial methods.

Abstract

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.