Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces

Abstract

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we compute the transition formulas for Donaldson polynomial invariants of rational surfaces. Part of the work is also obtained independently by Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.