Izzet Coskun, University of Illinois at Chicago
4:00pm, 25 marzo: Brill-Noether Theorems on moduli spaces of sheaves on rational surfaces.
Let X be a smooth, projective surface and let M be a moduli space of sheaves on X. After giving an introduction to the geometry of M, I will describe joint work with Jack Huizenga on understanding the cohomology of the general sheaf in M when X is a Hirzebruch surface. I will give applications to the classification of globally generated bundles and Ulrich bundles. If time permits, I will explain how to classify stable bundles on Hirzebruch surfaces.
10:00am, 26 marzo: Bridgeland stability and Brill-Noether theorems on moduli spaces of sheaves on K3 surfaces.
I will give a brief introduction to Bridgeland stability conditions. I will then describe joint work with Howard Nuer and Kota Yoshioka on determining the cohomology of the general sheaf on a moduli space of sheaves on a K3 surface.
4:00pm, 26 marzo: Informal conversation.
10:00am, 27 marzo: Ample and effective cones of moduli spaces of sheaves on the plane.
Following joint work with Jack Huizenga and Matthew Woolf, I will describe how to compute the ample and effective cones of moduli spaces of sheaves on the projective plane.
4:00pm,
27 marzo: Informal conversation.
Work under discussion
Brill-Noether Theorems and globally generated vector bundles on Hirzebruch surfaces (joint with Jack Huizenga):
(arXiv)
The effective cone of the moduli space of sheaves on the plane (joint with Jack Huizenga and Matthew Woolf):
(arXiv)
Construction of moduli spaces of curves
10:00am, 3 abril: Moduli stacks of curves with worse-than-nodal singularities.
I will describe natural variations of the Deligne-Mumford
stability condition on the moduli stack of all proper curves,
including pseudostability (Schubert), h-semistability (Hassett-Hyeon),
and alpha-stability (from joint work with Jarod Alper and David
Smyth), followed by a discussion of what aspects of stability give
rise to well-behaved moduli stacks.
10:00am, 4 abril: Construction of moduli spaces: techniques and examples.
This will be a crash course on general techniques for
constructing moduli spaces in algebraic geometry: GIT, Keel-Mori
theorem for Deligne-Mumford stacks, and its recent generalizations due
to Alper-Halpern-Leistner-Heinloth.
4:00pm, 4 abril: Informal conversation.
10:00am, 5 abril: Construction of moduli spaces II: the moduli spaces of
alpha-stable curves.
This lecture will be devoted to a proof of the existence
of projective moduli spaces of alpha-stable curves as defined in the
trilogy "Second flip in the Hassett-Keel program" (joint work with
Alper and Smyth).
4:00pm,
5 abril: Informal conversation.
Work under discussion