Main seminar (2:30-3:30)
Title: Stable sheaves in projective space
Abstract: Moduli spaces of vector bundles or more generally sheaves on algebraic varieties are usually badly behaved. As soon as the dimension of the variety is at least three, they satisfy Murphy's Law in algebraic geometry, i.e., all types of singularities can occur on them. In this talk, I will introduce a class of stable sheaves in three-dimensional projective space, whose moduli spaces are smooth and irreducible, contrary to the general picture. These spaces are closely related to the numerical study of Chern characters of semistable sheaves.
Title: Non-empty moduli spaces
Abstract: Moduli spaces are so ubiquitous in algebraic geometry that it might be surprising to find that non-emptiness can be a huge issue. Drezet and Le Potier managed to classify the numerical invariants of sheaves on the projective plane that can be parametrized in a reasonably behaved moduli space. Their result is unexpectedly complicated involving a fractal curve. In this talk, I will explain the necessary background to understand their result and give some hints to how it can be proved.