# Posters

## WAGS Spring 2023 @UW

Harpreet S. Bedi (Alfred U.)

Title: Modulo p equivalence of categories.

Abstract: In this manuscript/poster rational degree polynomial rings are constructed and it is shown that these rings are well behaved. The techniques of algebraic geometry can be applied to these rings to construct vector bundles and compute the Picard group. As an application, these rings are then used to construct an equivalence between category of rings of char 0 and char p by using a humble modulo p map.

Alvaro Cornejo (UKY)

Title: Algebra and Combinatorics of the Brion Map on Generalized Permutahedra

Abstract: Prior work has put a Hopf Monoid (and Hopf Algebra) structure on combinatorial objects, such as posets, and on a special family of polyhedra called Generalized Permutahedra. With this algebraic structure we can study maps that preserve these now algebraic objects. In particular, we studied the Brion map which takes in Generalized Permutahedra and maps these to some associated poset(s) coming from the vertices. We will explore this map more specifically on associahedra.

Louis Esser (UCLA)

Title: Automorphisms of Weighted Projective Hypersurfaces

Abstract: Automorphism groups of smooth hypersurfaces in projective space are well studied in algebraic geometry. We explore the more general setting of automorphism groups of quasismooth hypersurfaces in weighted projective space and answer the following questions: when are these groups linear? When are they finite, and if finite, how large can they get? What does the automorphism group of a very general hypersurface with given weights and degree look like? In each case, we generalize and strengthen the analogous results on ordinary projective hypersurfaces.

Fernando Figueroa Zamora (Princeton)

Title: Fundamental groups of low-dimensional log canonical singularities.

Abstract: Over the complex numbers, the fundamental group of a singularity can be defined as the fundamental group of a sufficiently small punctured neighborhood of an algebraic singularity. We study the fundamental groups of log canonical singularities of dimension at most 4. In dimension 3, we show that every surface group appears as the fundamental group of a 3-fold log canonical singularity. In contrast, we show that for $r \geq 2$ the free group on $r$ generators is not the fundamental group of a 3-dimensional lc singularity. In dimension 4, we show that the fundamental group of any 3-manifold smoothly embedded in $\mathbb{R}^4$ is the fundamental group of an lc singularity. In particular, every free group is the fundamental group of a log canonical singularity of dimension 4.

Javier González Anaya (UCR)

Title: The hidden structure of negative curves

Abstract: The problem of determining the MDS property for blowups of weighted projective planes (WPP) at a general point has received renewed interest because of the essential role it plays in Castravet and Tevelev’s proof that M_{0,n} is not a MDS for n>>1. Such a blowup is a MDS if and only if it contains a negative curve different from E and another curve disjoint from it. From a toric perspective, a WPP is defined by a rational plane triangle. We consider a parameter space of triangles and see how the negative curves and the MDS property vary within it. Using this approach we recover and expand most known results in the area, including examples having a semi-open effective cone. This is joint work with Jose Luis González and Kalle Karu.

Joseph Helfer (USC)

Title: Rational elliptic surfaces, Severi curves, and quasi-modular forms

Abstract: Joint with François Greer and John Sheridan. We study the Severi varieties of bisections B on rational elliptic surfaces: the closure of the locus of irreducible, rational curves linearly equivalent to B. We relate such curves to Noether-Lefschetz theory for K3 surfaces, which studies members of families of surfaces with jumping Picard rank. Using a connection between Noether-Lefschetz theory and modular forms, initiated by Maulik-Pandharipande (based on work of Borcherds and Kudla-Milson), we give a new upper bound on the geometric genus of the Severi variety (which is a curve in the case at hand).

Yifeng Huang (UBC)

Title: Punctual Quot scheme on cusp via Gröbner stratification

Abstract: (Joint with Ruofan Jiang) We prove a rationality result on the motive (in the Grothendieck ring of varieties) of the Quot scheme of points on the cusp singularity x^2=y^3, extending a well-known phenomenon for the Hilbert scheme of points on singular curves. Our method is based on stratification given by Gröbner theory. The essential combinatorial ingredient behind the rationality is a family of “spiral shifting” operators on {0,1,2,…}^d, originally developed by the authors to study the enumeratives of full-rank sublattices of Z^d.

Giovanni Inchiostro (UW)

Title: Degenerations of maps to algebraic stacks

Abstract: In this poster I will present some recent results regarding compactifications of morphisms from a family of curves over the punctured disc, to an algebraic stack admitting a projective good moduli space. It is based on a joint work with Andrea Di Lorenzo.

Seth Ireland (CSU)

Title: Strongly stable partitions and a bijection

Abstract: Artinian monomial ideals in d variables correspond to d-dimensional partitions. We define d-dimensional strongly stable partitions and show that they correspond to strongly stable ideals in d variables. There is a bijection between strongly stable partitions and totally symmetric partitions which preserves the side length of the minimal bounding box, which allows us to enumerate the strongly stable partitions in the case d=3.

Jae Hwang Lee (CSU)

Title: A Quantum H^*(G)-module via Quasimap Invariants

Abstract: For X a smooth variety or Deligne—Mumford stack, 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. For a GIT quotient V//G, the cohomology ring H^*(V//G) also has the structure of a H^*(G)-module. In this work, we use quasimap invariants with light points and a modified version of the WDVV equation to define a quantum deformation of this H^*(G)-module structure. Using localization, we explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the Batyrev ring when the target is a semipositive toric variety.

Peter McDonald (Utah)

Title: A new characterization of the multiplier ideal and klt singularities

Abstract: This project gives a new characterization of the multiplier of a scheme X by looking at images of maps R\pi_*\omega_Y=\pi_*\omega_Y\to O_X where \pi:Y\to X ranges over all regular alterations. As a corollary to this result, we give a derived splinter characterization of klt singularities akin to the characterization of rational singularities given by Bhatt and Kovács.

Harry Richman (Fred Hutch)

Title: Tropical weights of Weierstrass components

Abstract: For an algebraic curve of genus g, a Weierstrass point is a point where the canonical embedding of the curve in projective space is ramified. A classical theorem of algebraic geometry states that the total number of Weierstrass points, counted with appropriate weights, is g^3 - g. In the tropical world, it is natural to define Weierstrass points using analogues of divisors and linear systems, but the locus of Weierstrass points can generally be infinite. In recent work with Omid Amini and Lucas Gierczak, we show that we can associate a natural tropical weight to each connected component of the tropical Weierstrass locus, such that the total weight is g^2 - 1.

Shravan Patankar (UIC)

Title: Coherence of absolute integral closures

Abstract: In spite of being large and non noetherian absolute integral closures are of tremendous importance in commutative algebra and algebraic geometry. We show that they are not even coherent. In positive characteristic this statement is due to Aberbach and Hochster. Our new techniques prove the statement in equicharacteristic zero and give an astonishingly simple proof in mixed characteristic. Previous proofs of the mixed characteristic statements used complicated machinery such as perfectoid spaces whereas our approach is elementary and extends to new dimensions.

Andrew Tawfeek (UW)

Title: A Tropical Framework for Using Porteous' Formula

Abstract: We present preliminary research in developing the tools necessary to study degeneracy loci of tropical vector bundles, and in particular, applying Porteous’ formula. This largely centered around further developing the theory of characteristic classes. This is a work in progress.

Joshua Turner (Davis)

Title: Homology of Affine Springer Fibers

Abstract: Affine Springer fibers are geometric spaces often studied in geometric representation theory. They have strong connections to the Hillbert scheme of singular curves, as well as knot homology. In this poster, we will show some examples of what these affine Springer fibers look like and how to calculate their equivariant Borel-Moore homology.

Ming Zhang (UCSD)

Title: Orbifold quantum K-theory

Abstract: Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov-Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this poster, I will define a quantum K-ring that specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. I will also explicitly compute the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This is joint work with Yang Zhou.