Generalized Divisors on DMH Stacks
Minghua Dou
TL;DR
The paper addresses extending generalized divisors from schemes to DMH stacks, defined via the Hartshorne $G_{1}+S_{2}$ condition on étale charts. It shows that defining divisors via fractional ideals of the total quotient sheaf on stacks is problematic, and instead introduces a robust framework using reflexive coherent $\mathcal{O}_{\mathcal{X}}$-modules locally free of rank $1$ at generic points, with duals realizing the divisor. This approach recovers Hartshorne's notion when the stack is a scheme, and enables a notion of linear systems and divisors without embedded points on DMH stacks, while outlining open questions on sums, degrees, and group structures. This work lays groundwork for divisor theory on moduli stacks and informs future developments in algebraic geometry of stacks.
Abstract
Hartshorne developed a theory of generalized divisors on Gorenstein schemes to characterize codimension-one closed subschemes without embedded points. Generalized divisors can be viewed as a generalization of Weil divisors to non-normal schemes. The purpose of this paper is to extend generalized divisors on schemes to DMH stacks, where DMH stacks are Deligne-Mumford stacks satisfying specific conditions. We provide a detailed proof of the properties of total quotient sheaves under étale morphisms, thereby demonstrating the difficulty of directly defining generalized divisors on DMH stacks through fractional ideals of the total quotient sheaf. Instead, we propose to define generalized divisors on DMH stacks using reflexive coherent sheaves that are locally free of rank one at generic points. Furthermore, we rigorously establish the rationality of this definition on stacks.