## Moduli spaces of sheaves in algebraic geometry

### Prof. Alexander Tikhomirov

Number of cycles: 2

Second semester

Contents:

- Preliminaries and projective morphisms.
- Flatness and base change.
- Hypersurfaces in the projective space.
- Representable functors.
- Grassmannians.
- Flag varieties.
- Relative flag varieties.
- Hilbert schemes and Quot schemes.
- Tangent spaces to Quot schemes.
- Examples of Hilbert schemes and Quot schemes: the Hilbert scheme Hilb(P
^{2}, 2m+2) and the Quot scheme Quot(2O_{P2}, 2m+2).. - Semistable sheaves.
- The Serre construction of semistable sheaves.
- Harder-Narashiman filtration, Jordan-Hölder filtration and S-equivalence.
- The moduli functor.
- Group actions.
- The GIT construction of moduli spaces.
- The GIT construction - proofs.
- Examples of moduli spaces: moduli spaces of semistable sheaves on the projective plane.
- Mathematical instantons: the monad description.
- Moduli spaces of mathematical instantons.