Posters

Joshua Enwright: Log Calabi-Yau pairs of birational complexity zero. 

A log Calabi-Yau pair has birational complexity zero if it admits a crepant birational toric model. We introduce invariants motivated by toric geometry and combinatorics to study the geometry of these pairs. We prove basic structural results and explore their boundedness properties.

This is joint work with Fernando Figueroa and Joaquín Moraga.

Louis Esser: The Dual Complex of a G-variety 

We introduce a new invariant of G-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant of G-varieties. As an application, we demonstrate the non-linearizability of certain large abelian group actions on smooth hypersurfaces in projective space of any dimension and degree at least 3. 

Fernando Figueroa: Fundamental groups of low coregularity Calabi-Yau type pairs 

It was shown by Braun that the fundamental group of the smooth locus of Fano type pairs is finite. This is false for Calabi-Yau type pairs, even in dimension one. The purpose of this poster is to present how the fundamental group behaves in low coregularity Calabi-Yau type pairs.

The main theorems are the finiteness of the fundamental group in coregularity 0 and the similarities between low coregularity and low dimension.

This is based on joint work with Lukas Braun. 

Casey Hill: The Geometry of SL_4-Conformal Blocks and the Theta Divisor 

Conformal blocks are objects from mathematical physics which appear naturally as the spaces of global sections of line bundles on the moduli of vector bundles on smooth curves. We will discuss how to determine a presentation for the algebra of SL_4-conformal blocks and its Theta divisor. We will also discuss some of the geometric data that can be uncovered from their tropical varieties, which allows us to compute their toric degenerations. 

Daigo Ito: Gluing of Fourier-Mukai partners using tensor triangulated geometry 

We initiate studies of the relationship between birational geometry of smooth varieties and tensor structures of their derived categories. A crucial insight from tensor triangulated geometry is that we can view a variety as a specific tensor structure on its derived category and an upshot is that closer observations on such tensor structures allow us to glue its Fourier-Mukai partners to construct a smooth scheme locally of finite type, which we call the Fourier-Mukai locus. The geometry of the locus reflects geometric properties of the Fourier-Mukai partners in various contexts, such as abelian, toric, and Fano varieties, as well as birational operations among them such as flops. We further investigate relations with the DK hypothesis and methods to purely categorically characterize the locus. 

Soyeon Kim: Frozen variables for open Richardson varieties

In an effort to suggest a non recursive formula for R-polynomials,  open Richardson varieties were invented in 1979 by Kazhdan and Lusztig. In the years since, many surprising connections have emerged between open Richardson varieties and topics far from those motivating Kazhdan and Lusztig. One of the most noteworthy discoveries is that the coordinate ring of open Richardson varieties have a cluster algebra structure. In this poster, we address how to relate R-polynomials with cluster algebra structures of open Richardson varieties. 

Jae Hwang Lee : A Quantum H*(T)-module via Quasimap Invariants 

For X a smooth projective variety, the quantum cohomology ring QH*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov--Witten invariants. When X is toric with  geometric quotient description V// T, the cohomology ring H*(V//T) also has the structure of a H*(T)-module. In this paper, we introduce a new deformation of the cohomology of X using quasimap invariants with a light point. This defines a quantum H*(T)-module structure on H*(X) through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety. 

Yuze Luan :  Irreducible components of Hilbert scheme of points on non-reduced curves

We classify the irreducible components of the Hilbert scheme of n points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of n; all have dimension n; and their multiplicities are given as a polynomial of the parts of the corresponding partitions.

Flora Poon: Kuga-Satake varieties of families of K3 surfaces of Picard rank 14 

The classical Kuga-Satake construction takes a K3 surface to an abelian variety. We lift the Kuga-Satake construction to the level of moduli spaces for K3 surfaces polarised by a rank 14 lattice: It determines a map between the moduli space of K3 surfaces and that of abelian varieties with appropriate structures. This is due to the coincidence of a type IV_6 and a type II_4  Hermitian symmetric domains induced from an exceptional Lie algebra isomorphism. Furthermore, we realise the construction using the MAGMA computational algebra system for a few specific families and study some special loci. 

Debaditya Raychaudhury : On the singularities of secant varieties 

In this work, we study the singularities of secant varieties of smooth projective varieties when the embedding line bundle is sufficiently positive. We give a necessary and sufficient condition for these to have p-Du Bois singularities. In addition, we show that the singularities of these varieties are never higher rational. From our results, we deduce several consequences, including a Kodaira-Akizuki-Nakano type vanishing result for the reflexive differential forms of the secant varieties. Work in collaboration with S. Olano and L. Song. 

Pijush Pratim Sarmah: Jacobians of Curves in Abelian Surfaces 

Every curve has an abelian variety associated with it, called the Jacobian. Poincaré's total reducibility theorem states that any abelian variety is isogenuous to a product of simple abelian varieties. We are interested to know this decomposition for Jacobians of smooth curves in abelian surfaces. Using Kani and Rosen's strikingly simple yet powerful theorem that relates subgroups of automorphism groups with isogeny relations on Jacobians, we will decompose Jacobians of certain curves coming from linear systems of polarizations on abelian surfaces. We will also look at when these curves can be hyperelliptic or have elliptic factors in the isogeny classes of their Jacobians.  

Jose Yanez: Polarized endomorphisms of Fano varieties with complements. 

An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. It is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. We consider the case when X is a Fano variety admitting a 1-complement, meaning that there exists an effective divisor B such that (X,B) is log Calabi-Yau and K_X + B ~ 0. Then, if a polarized endomorphism of X preserves the complement structure, we prove that (X,B) is a finite quotient of a toric log Calabi-Yau pair.

 This is joint work with Joaquin Moraga and Wern Yeong.