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

摘要

Let $X$ be a complex projective manifold. Fix two ample line bundles $H_0$and $H_1$ on $X$. It is the aim of this note to study the variation of themoduli spaces of Gieseker semistable sheaves for polarizations lying in thecone spanned by $H_0$ and $H_1$. We attempt a new definition of walls whichnaturally describes the behaviour of Gieseker semistability. By means of anexample, we establish the possibility of non-rational walls which is asubstantially new phenomenon compared to the surface case. Using the approachof Ellingsrud and Goettsche via parabolic sheaves, we were able to show thatthe moduli spaces undergo a sequence of GIT flips while passing a rationalwall.