Integration on Artin toric stacks and Euler characteristics

TL;DR

This paper extends intersection theory and defines Euler characteristics for Artin toric stacks with complete underlying varieties, providing explicit formulas in three dimensions, thus broadening the scope of algebraic geometry tools.

Contribution

It introduces a new definition for the degree of cycles on Artin toric stacks with non-quasi-finite diagonals and extends Euler characteristic concepts to these stacks.

Findings

Defined the degree of cycles on Artin toric stacks with complete varieties.

Extended the orbifold Euler characteristic to Artin toric stacks.

Provided explicit combinatorial formulas for 3-dimensional cases.

Abstract

There is a well developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space - extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks.