On finite generation of the section ring of the determinant of cohomology line bundle

TL;DR

This paper proves the finite generation of the section ring associated with the determinant of cohomology line bundle on the moduli space of SL(r)-bundles over a stable curve, with three applications.

Contribution

It establishes the finite generation of the section ring for the determinant of cohomology line bundle on the moduli space of SL(r)-bundles over stable curves, a key step in understanding their algebraic structure.

Findings

Section ring is finitely generated for stable curves of genus ≥ 2.

Provides three applications related to moduli spaces and line bundles.

Advances the understanding of algebraic properties of moduli of vector bundles.

Abstract

For $C$ a stable curve of arithmetic genus $g\ge 2$, and $\mathcal{D}$ the determinant of cohomology line bundle on $\operatorname{Bun}_{\operatorname{SL}(r)}(C)$, we show the section ring for the pair $(\operatorname{Bun}_{\operatorname{SL}(r)}(C), \mathcal{D})$ is finitely generated. Three applications are given.