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

1994/10/15 by Matsuki, K., Wentworth, R.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.alg-geom/9410016

Abstract

We study the behavior of the Gieseker space of semistable torsion-free sheaves of rank r and fixed c₁, c₂ 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).

Related

• Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases
• Change of polarization for moduli of sheaves on surfaces as Bridgeland wall-crossing
• Irreducibility of the moduli space of vector bundles on surfaces and Brill-Noether theory on singular curves
• Flips and variation of moduli schemes of sheaves on a surface
• Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank