Globally F-regular type of moduli spaces and Verlinde formula

TL;DR

This paper establishes that certain moduli spaces are of globally F-regular type and derives a Verlinde formula for their theta function spaces, advancing understanding of their geometric and algebraic properties.

Contribution

It proves the globally F-regular type of moduli spaces and derives an explicit Verlinde formula for theta function dimensions.

Findings

Moduli spaces are globally F-regular type.

Derived vanishing theorems on singular curves.

Established recurrence relations and explicit Verlinde formula.

Abstract

We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions.