# 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: Genus two reduced quasi-map invariants for CY3 complete intersections
Abstract: In joint work with Jeongseok Oh and M.-L. Li, we define genus two reduced quasi-map invariants for Calabi-Yau 3-folds embedded in projective space or product of complete intersections, and study standard versus reduced formula as an analogue of the genus 1 case proved by Zinger.

For the definition of reduced invariants and proof of the formula, we use desingularization of genus two quasi-map space, which is a simple analogue of the desingularization of the genus two stable map space constructed by Hu-Li-Niu.

We describe this desingularization and its local chart, local equation explicitly and using this, we explain how we can split virtual cycle to a reduced part (on main component) and boundary part. If time is allowed, I will also explain how we can compute boundary virtual cycle to obtain the formula.
Notes

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

Title: The Freed-Hopkins-Teleman theorem
Abstract: For a given compact Lie group G, the Freed-Hopkins-Teleman theorem states that the level k fusion ring of G is isomorphic to the twisted equivariant K-theory of G, with twisting determined by the level k. In this talk, we will attempt to outline some of the ideas and constructions involved in the proof of this result, in the special case where G is 1-connected and simple.

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

Title: Virasoro constraints in quantum singularity theory
Abstract: In this talk, we introduce Virasoro operators in quantum singularity theories for nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. Using Givental's quantization formula of quadratic Hamiltonians, these operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the genus g generating functions arise in quantum singularity theories are annihilated by the Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and LG/CY correspondence. This talk is based on work joint with Weiqiang He.

• Kirti Joshi (University of Arizona)
Thursday, February 11, 2020, 3:00 PM

Title: On the construction of weakly Ulrich bundles
Abstract: A well-known conjecture of Eisenbud, Schreyer and Weyman suggests that any projective variety carries an Ulrich bundle and hence also weakly Ulrich bundle. This is presently known only in a handful of cases. I will explain this conjecture and explain some of my recent results in the direction of this conjecture which provide a construction of weakly Ulrich bundles in a number of new cases. These results also imply that Chow form of these varieties is the support of a single intrinsic determinantal equation.