Stable pairs, linear systems and the Verlinde formula

TL;DR

This paper investigates the moduli space of rank 2 vector bundle pairs with sections over a smooth curve, revealing flip transformations and deriving key formulas like the Verlinde formula through geometric and algebraic methods.

Contribution

It introduces a new stability parameter framework for moduli of pairs, demonstrating flip transformations and deriving the Verlinde formula from geometric principles.

Findings

Moduli space undergoes a sequence of flips as stability varies.

Proves the Harder-Narasimhan formula for rank 2 bundles.

Derives the SU(2) Verlinde formula from geometric analysis.

Abstract

We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.