Mathematics

The modern Geometric Invariant Theory (GIT) is one of the main tools used in constructing various moduli spaces of (semi)stable coherent sheaves on projective schemes. The aim of this lecture course is to provide some basic techniques and constructions used in the application of GIT to the description of moduli spaces of sheaves. We suppose to cover the following topics: coherent sheaves on projective schemes, flatness and morphisms, Quot-functor, Quot-schemes and Hilbert schemes, semistability of sheaves, representable functors, the moduli functor, group actions, categorical and geometric quotients, GIT, the construction of moduli spaces of semistable sheaves via GIT. As an application of the above results we consider a number basic examples of moduli spaces such as moduli of zero-dimensional sheaves, moduli spaces of stable vector bundles on projective curves, moduli spaces of semistable sheaves on projective plane and projective 3-space, their geography and geometry, and related questions.