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

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

1999-05-19

arxiv.org/abs/math/9905110v2

Mathematics

Algebraic Geometry

10pp, AmSLaTeX (a4paper); Revised Version (some typos removed, Ex.1.1.5 completed)

Scientific paper

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 the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by $H_0$ and $H_1$. 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 of Ellingsrud and Goettsche via parabolic sheaves, we were able to show that the moduli spaces undergo a sequence of GIT flips while passing a rational wall.

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