Mumford-Thaddeus Principle on the Moduli Space of Vector Bundles on an Algebraic Surface

TL;DR

This paper investigates how the moduli space of semistable sheaves on a smooth projective surface changes as the polarization varies, revealing a chamber structure and flip transformations at wall crossings.

Contribution

It introduces a new chamber structure for the ample cone and describes the wall-crossing behavior of the moduli space via flips, extending previous methods with a rationally twisted approach.

Findings

Ample cone admits a locally finite chamber structure.

Wall crossings induce finite sequences of flips in the moduli space.

The method generalizes previous results with a new rational twist.

Abstract

We study the behavior of the Gieseker space of semistable torsion-free sheaves of rank r and fixed c_1, c_2 on a non-singular projective surface as the polarization varies. It is shown that the ample cone admits a locally finite chamber structure, and that passing a wall adjacent to a pair of chambers has the effect of modifying the moduli space by a (finite) sequence of flips of the type studied by Thaddeus. The key steps are a modification of Simpson's method and the introduction of a "rationally twisted" moduli space. The result is more general but less explicit than the recent work of Ellingsrud-Goettsche (alg-geom/9410005) and Friedman-Qin (alg-geom/9410007).