Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases

Abstract

Let X be a projective manifold over $\mathbb{C}$ . Fix two ample line bundles H0 and H1 on X. It is the aim of this note to study the variation of the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by H0 and H1. We attempt a new definition of walls which naturally describes the behaviour of Gieseker semistability. By means of an example, we establish the possibility of non-rational walls which is a substantially new phenomenon compared to the surface case. Using the approach by Ellingsrud and Göttsche via parabolic sheaves, we were able to show that the moduli spaces undergo a sequence of GIT flips while passing a rational wall. We hope that our results will be helpful in the study of the birational geometry of moduli spaces over higher dimensional bases.