FRG Workshop on Moduli Spaces and Stability
December 7-10, 2020
on Zoom


Organizers: İzzet Coşkun (U. Illinois at Chicago), Jack Huizenga (Penn State U.), John Kopper (Penn State U.), Emanuele Macri (U. Paris-Saclay), Alina Marian (Northeastern U.), Aaron Bertram (U. Utah)

Registration: Registration is free and open to everyone. If you are planning to attend, please register by December 1, 2020 by filling out the registration form here. We will send the zoom links to registered participants on December 4.

Schedule (All times are Chicago/Central time):
Monday Dec 7, 2020 Here is a link for the recordings: Recordings of the first day
9:00-10:00 Alexander Kuznetsov
10:15-11:00 Pierrick Bousseau Slides are available here
11:10-11:35 Dmitrii Pedchenko Slides are available here
11:45-12:10 Hannah Larson

Tuesday Dec 8, 2020 Here is a link for the recordings: Recordings of the second day:
9:00-10:00 Alexander Kuznetsov
10:15-11:00 Hülya Argüz
11:15-12:00 Dmitrii Pirozhkov Slides are available here

Wednesday Dec 9, 2020 Here is a link for the recordings: Recordings of the third day
10:00-11:00 Alexander Kuznetsov
11:15-12:00 John Kopper

Thursday Dec 10, 2020 Here is the link for the recordings: Recordings of the fourth day :
9:00-10:00 Alexander Kuznetsov
10:15-11:00 Soheyla Feyzbakhsh
11:15-12:00 Naoki Koseki

Titles and Abstracts:

Hülya Argüz:
Title: Enumerating punctured log Gromov--Witten invariants from wall-crossing
Abstract: Punctured log Gromov-Witten invariants of Abramovich-Chen-Gross-Siebert are obtained by counting stable maps with prescribed tangency conditions (which are allowed to be negative) relative to a not necessarily smooth divisor. We describe an algorithmic method to compute punctured log Gromov-Witten invariants of log Calabi-Yau varieties, which are obtained by blowing-up of toric varieties along hypersurfaces on the toric boundary. For this we use tropical geometry and wall-crossing computations. This is joint work with Mark Gross (arxiv:2007.08347).

Pierrick Bousseau:
Title: A scattering diagram for coherent sheaves on the projective plane
Abstract: I will describe a new algorithm computing intersection Betti numbers of moduli spaces of semistable coherent sheaves on the projective plane. The algorithm takes the form of a scattering diagram embedded in the space of Bridgeland stability conditions on the derived category of coherent sheaves.

Soheyla Feyzbakhsh:
Title: An application of a Bogomolov-Gieseker type inequality to counting invariants
Abstract: In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas.

John Kopper:
Title: The cohomology of general tensor products of vector bundles on the projective plane
Abstract: For two general stable vector bundles V and W on the plane, we explain how to compute the cohomology of their tensor product when the Chern character of W is assumed to be sufficiently divisible. In the special case that W is an exceptional bundle, we show that the tensor product has at most one nonzero cohomology group, generalizing results of Drézet and Göttsche and Hirschowitz. When neither V nor W is exceptional, this can fail, i.e., the general tensor product can have two nonzero cohomology groups. The proof illuminates some interesting aspects of the birational geometry of moduli spaces of stable sheaves. This is joint work with Izzet Coskun and Jack Huizenga.

Naoki Koseki:
Title: Bridgeland stability on algebraic surfaces in positive characteristic
Abstract: I will explain my recent work constructing Bridgeland stability conditions on all smooth projective surfaces in positive characteristic. The key is proving a suitable quadratic inequality of Bogomolov-Gieseker type.

Alexander Kuznetsov:
Title: Exceptional collections on homogeneous varieties of simple algebraic groups
Abstract: In this minicourse I will review what is known and what is expected about the structure of the derived categories of coherent sheaves on compact homogeneous spaces of simple algebraic groups, starting from classical results of Beilinson and Kapranov, and hopefully going up to the recent progress in the field.

Hannah Larson:
Title: Brill--Noether theory over the Hurwitz space
Abstract: Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting.

Dmitrii Pedchenko:
Title: The Picard group of the moduli space of bundles on a quadric surface
Abstract: In this talk, I will discuss the computation of the Picard group of the moduli space of bundles on a smooth quadric surface. I will focus especially on moduli spaces of bundles of small discriminant, where we observe new and interesting behavior. I will outline methods for constructing certain resolutions for semistable bundles on the quadric and applying techniques of geometric invariant theory to the resulting families of bundles to study the moduli space. The talk is based on

Dmitrii Pirozhkov:
Title: Admissible subcategories of del Pezzo surfaces
Abstract: Semiorthogonal decompositions help up understand the structure of derived categories of coherent sheaves on algebraic varieties. There are many examples of them, but not so much is known about the set of all possible SODs, even for varieties as simple as projective spaces. I will explain the proof of the full classification of decompositions for D(P^2) and, if time permits, also some partial results about derived categories of sheaves on del Pezzo surfaces, notably the non-existence of phantom subcategories in them.

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.