Mathematical Physics Sector

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SISSA — International School for Advanced Studies Phonebook Webmail Library catalogue

Via Bonomea 265 — 34136 Trieste — Italy 45.679538°N, 13.774999°E

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(P², 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.