Algebraic Geometry Seminar

Time: Thursday 3-4pm (Fall 20)
Location: Zoom: https://arizona.zoom.us/j/95687864835
Calendar: http://websites.math.arizona.edu/ag-seminar/

Talks

• Darlayne Addabbo (University of Arizona)
Thursday, September 17, 2020, 3:00 PM

Title: Zhu algebras for vertex operator algebras
Abstract: Given a vertex operator algebra, $V$, there is a family of associative algebras, $A_n(V)$, $n\in \mathbb{N}$, known as Zhu algebras, which can be used to study the representation theory of $V$. In this talk, I will define these Zhu algebras and discuss motivation for their study. I will then discuss techniques used in determining their structure and give an example clarifying the necessity of certain conditions in defining the Zhu algebras, $A_n(V)$ for $n>0$. I will not assume prior knowledge of vertex operator algebras. (This is joint work with Katrina Barron.)

• David Jensen (University of Kentucky)
Thursday, September 24, 2020, 3:00 PM

Title: Hurwitz-Brill-Noether Theory
Abstract: The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether Theorem, which says that the space of linear systems with given discrete invariants on a general curve has the expected dimension. In this talk, we will discuss a generalization of this theorem to curves that are general in the Hurwitz space, rather than in the moduli space of curves. This is joint work with Kaelin Cook-Powell.
Notes

• Anand Patel (Oklahoma State University)
Thursday, October 1, 2020, 3:00 PM

Title: How Some Constructions in Projective Geometry Vary
Abstract: Many standard constructions in the study of algebraic varieties in projective space vary with natural inputs: hyperplane sections vary with the hyperplane, ramification loci vary with the center of projection, and so on. I hope to explain why an in-depth look at this variation opens up a fertile domain of inquiry, all the while reporting on relevant joint work with A. Deopurkar (ANU), E. Duryev (Univ. Paris Diderot), D. Tseng (MIT), and E. Riedl (Notre Dame).
Notes

• Francesca Carocci (University of Edinburgh)
Friday, October 9, 2020, 1:00 PM

Title: A modular smooth compactification of genus 2 curves in projective spaces
Abstract: Moduli spaces of stable maps in genus bigger than zero include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe modularly.

On the other hand, after the work of Li-Vakil-Zinger and Ranganathan-Santos-Parker-Wise in genus one, we know that points in the boundary of the main component correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities on which the map is less degenerate. Morally, such a Gorenstein curve is obtained by contracting any higher genus sub-curve on which the map is constant.

The question becomes how to construct such a universal family of Gorenstein curves to then single out the (resolution) of the main component of maps imposing the factorization property. In joint work with L. Battistella, we construct one such family in genus two over a logarithmic modification of the space of admissible covers, and consequently obtain the desired smooth compactification of genus 2 curves in projective spaces.
Notes

• Yunfeng Jiang (University of Kansas)
Thursday, October 29, 2020, 3:00 PM

Title: Vafa-Witten invariants and S-duality
Abstract: For a real four manifold M, the S-duality conjecture of Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons on M to the Langlands dual gauge group SU(r)/Z_r-invariants counting SU(r)/Z_r-instantons on M; and both of the invariants satisfy modularity properties. This is a generalization of electro-magnetic duality in physics. On mathematics side the SU(r)-Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth complex projective surface underlying M. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Z_r-Vafa-Witten invariants, and provide the proposal to prove the S-duality conjecture of Vafa-Witten for algebraic surfaces. A particular case of K3 surface is proved under this proposal.

• Sanghyeon Lee (Korea Institute for Advanced Study)
Thursday, November 5, 2020, 3:00 PM

Title: TBD Abstract: TBD

• Ahmed Zerouali (University of Arizona)
Thursday, November 12, 2020, 3:00 PM

Title: TBD Abstract: TBD

• Yefeng Shen (University of Oregon)
Thursday, November 19, 2020, 3:00 PM

Title: TBD Abstract: TBD