FRG Lectures on Moduli Spaces of Sheaves on Surfaces
by Carlos Simpson
June 10-11, 2021
on Zoom

Speaker: Carlos Simpson (CNRS, Université Côte d'Azur, visiting IAS Princeton)

Organizers: İzzet Coşkun (UIC), Jack Huizenga (Penn State U.), John Kopper (Penn State U.), Geoffrey Smith (UIC)

Registration: Registration is free and open to everyone. If you are planning to attend, please register by June 8, 2021 by filling out the registration form here. We will send the zoom links to registered participants on June 9 in the evening.

Schedule (All times are Chicago/Central time):
Here are the lecture notes (pdf)
Thursday June 10, 2021: The recordings are (here)
1:00 - 1:50 pm Lecture 1: Moduli spaces of sheaves--an overview of the geography
Abstract: In this first talk we will look more globally at the properties of moduli spaces of vector bundles and sheaves on various kinds of varieties, then specializing to the case of surfaces and in particular the case of rank 2 bundles on quintic hypersurfaces in P^3.

2:00 - 2:50 pm Lecture 2: Constructions and local properties: the Cayley-Bacharach condition
Abstract: In the second talk, we look at the Serre construction of rank 2 vector bundles using the Cayley-Bacharach condition on zero-dimensional subschemes. Topics include the local deformation theory, how it interacts with the Serre construction, and the interpretation of co-obstructions as Higgs fields.

Friday June 11, 2021: The recordings are (here)
1:00 - 1:50 pm Lecture 3: Ascending induction and O'Grady's method
Abstract: In the third talk, we introduce O'Grady's method of deforming to the boundary by creating deformations from bundles to torsion-free sheaves. Combining that with explicit constructions for low values of c_2, we look at the implications for the how the collection of moduli spaces fits together as c_2 increases.

2:00 - 2:50 pm Lecture 4: Structure of the moduli spaces and their boundaries
Abstract: The fourth talk combines the previous strands to obtain the picture of the moduli spaces of rank 2 bundles and sheaves of odd degree on a quintic hypersurface. Further questions are explored.

We are grateful for the generous support of the University of Illinois at Chicago, Penn State University and the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.