# Combinatorial AG Seminar

Welcome to the homepage for the Combinatorial Algebraic Geometry Seminar!

This seminar was organized by Andrew Tawfeek and Cameron Wright at the University of Washington.

It will resume in the Autumn 2024 quarter.

# Winter 2023 Talks

## Friday, January 20th, 2023

### Title: Algebraic Curves, Dual Graphs, and Jacobians Thereof

Speaker: Cameron Wright

Abstract: Algebraic curves are a central class of objects in algebraic geometry. Smooth curves in particular are a well-understood class of curves, and possess a robust moduli theory. On the other hand, curves with (nodal) singularities are somewhat less well-behaved; still, these curves feature prominently in modern algebraic geometry, in particular in the context of the Deligne-Mumford compactification of the moduli space of smooth curves. As such, it is of great interest to understand curves with these mild singularities. Over the past century, it became clear to algebraic geometers that much could be gained from studying combinatorial objects derived from these curves. In this talk, we survey this combinatorial approach and, if time permits, compare the construction of Jacobians in both the algebraic and combinatorial settings.

## Friday, January 27th, 2023

### Title: Tropical Ceresa cycles

Speaker: Caelan Ritter

Abstract: The Ceresa cycle C - C^{-} of a smooth algebraic curve C is a tautological algebraic cycle contained in the Jacobian J(C). It is homologically trivial, but Ceresa showed that if C is very general of genus at least 3, then it is not algebraically trivial. It is in some sense the simplest algebraic cycle satisfying these properties, leading to applications for the étale fundamental group, intersection theory, and the theory of heights. We will discuss the extent to which the Ceresa cycle and the proof of non-triviality carry over to the tropical setting. Along the way, we will introduce important tools in the study of rational polyhedral spaces, namely, tropical cycles and homology.

## Friday, February 3rd, 2023

### Title: The Virtual Euler Characteristic for Binary Matroids

Speaker: Andrew Tawfeek

Abstract: In the graph chain complex work of Kontsevich, he computed the graphic orbifold Euler characteristic and showed it may be fascinatingly expressed through Bernoulli numbers. Inspired by this, Madeline Brandt, Juliette Bruce, and Daniel Corey in arXiv: 2301.10108 define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank r, then prove this similarly expression is possible over the finite field F_2 (i.e. binary matroids). Additionally, they apply their methods to craft recursive formulas for subsets of the Grassmannian in the Grothendieck ring of varieties. In the talk, we briefly overview the combinatorial and algebro-geometric aspects of matroids and the Grassmannian before then delving into a discussion of their methods. We conclude the talk with various future directions one could take their work as well.

## Friday, February 10th, 2023

### Title: Weierstrass Points on Algebraic Curves and Tropical Curves

Speaker: Harry Richman

Abstract: An algebraic curve of high genus can be a difficult landscape to navigate. In such a situation, we would like a set of well-placed landmarks as a guide. This is what Weierstrass points provide for us. We will cover the definition of Weierstrass points, and overview results that quantify in what sense they are "well-placed". We will also mention some applications in algebraic geometry and tropical geometry, in particular:

When does an algebraic / tropical curve have finitely many automorphisms?

When is the moduli space of algebraic / tropical curves "general type"?

## Friday, February 24th, 2023

### Title: Compactifications of Jacobians of Nodal Curves

Speaker: Cameron Wright

Abstract: Jacobians of smooth curves are well-studied classical objects which have been studied since the work of Riemann. In the time since then, there have been many generalizations of this theory, including in particular constructions of Jacobians for curves with nodal singularities. These Jacobians are typically not projective and, as such, much work has been invested to compactify these objects. In this talk we study a family of compactifications of Jacobians of nodal curves due to Oda and Seshadri. These constructions rely on combinatorial and convex-geometric machinery which is interesting in its own right. Here, we survey both the algebro-geometric and the convex-geometric constructions needed to appreciate Oda-Seshadri's results.

## Friday, March 3rd, 2023

### Title: Tropical Vector Bundles and their Chern Classes

Speaker: Andrew Tawfeek

Abstract: We introduce the notion of a tropical vector bundle (of rank n) over a tropical cycle X, which is equivalently interpretable as a polyhedral complex living "above" X or GL_n-torsor. Primarily relying on the more-accessible combinatorial definition, we go on to look at morphisms, rational sections, and pull-backs of these bundles. After some examples, we will discuss how one can take local intersection products and construct Chern classes -- concluding the talk with a classification of all vector bundles on a (tropical) elliptic curve up to isomorphism, which will (surprisingly!) coincide with Atiyah's 1957 result in classical algebraic geometry.

## Friday, March 10th, 2023

### Title: Pseudo-Divisors and Polyhedral Connections

Speaker: Natasha Crepeau

Abstract: Divisors on graphs foster a rich connection between algebraic geometry and combinatorics, especially connections between the moduli theory of algebraic curves to graph theory and polyhedral geometry. In this talk we will walk through results presented in A universal tropical Jacobian over M_g^trop by Abreu, Andria, Pacini, and Taboada: we'll start with defining relevant divisors on tropical curves and graphs, introduce some stability conditions and special polytopes, and conclude with a construction of a universal tropical Jacobian.

# Spring 2023 Talks

## Monday, April 10th, 2023

### Title: Matrix-tree theorem: The Prym Case

Speaker: Junaid Hassan

Abstract: We review the classical matrix-tree theorem, and then discuss a generalization in the case of double covers: known as the Prym case. This talk is based on Y. Len, D. Zakharov [arxiv:2012.15235] and a recent continuing work by A. Ghosh, D. Zakharov [arxiv:2303.03904].

## Monday, April 17th, 2023

### Title: Mixed Volumes of Normal Complexes

Speaker: Lauren Nowak

Abstract: The establishment of the log-concavity of characteristic polynomials of matroids was a result that took decades to confirm and connected ideas from combinatorics, algebra, and algebraic geometry. In this talk, we will be exploring how this result can be achieved from the perspective of volume theory and polyhedral geometry. In particular, we will explore the conditions under which the mixed volumes of a normal complex - an orthogonal truncation of a polyhedral fan - satisfy the Alexandrov-Fenchel inequalities. By then specializing to Bergman fans of matroids, we give a new proof of the log-concavity of characteristic polynomials of matroids.

## Monday, May 1st, 2023

### Title: Vector bundles as multidivisors on metric graphs

Speaker: Andrew Tawfeek

Abstract: We introduce and discuss the notion of a (tropical) vector bundle on a metric graph (equivalently, an abstract tropical curve). We show that unlike the classical setting, we can express any vector bundle as the pushforward of a line bundle on a free cover of our graph. This lends itself to a natural notion of "multidivisors": a tropical incarnation of Weil's notion of a matrix divisor from Généralisation des fonctions abéliennes. We then use these notions to comment on the moduli space of vector bundles on a tropical curve, and prove analogues of the Narasimhan-Seshadri and Weil-Riemann-Roch theorems.

## Monday, May 22nd, 2023

### Special Week! Two general exams will take place!

Please note the times and rooms.

### Title: Pure Dimensionality problem for Pryms

Speaker: Junaid Hassan

Location: Smith Hall (SMI) 309

Time: 11:00 AM - 12:00 PM

Abstract: In this talk we explore double covers on graphs. We study the generalizations of break and semi-break divisors on double covers known as odd genus one decompositions or relative spanning trees. We discuss how they help us formulate and analyze pure dimensional problems on the Prym Variety associated with a metric graph. We also explore signed graphs and their relation to double covers.

### Title: Combinatorics of Compactified Jacobians

Speaker: Cameron Wright

Location: Thomson Hall (THO) 211

Time: 3:00 PM - 4:00PM

Abstract: The Jacobian of a smooth projective curve is a classical and well-studied object; it is an abelian variety which is known to serve as a moduli space for degree-0 line bundles on the curve. In the case that the curve possesses nodal singularities, the situation becomes more convoluted: the Jacobian of such curves fails to be projective. As such, many compactifications have been proposed over the years which have various desirable properties. One such family was constructed by Oda and Seshadri in 1979. The compactifications in this family are toroidal, and so admit a sophisticated description in terms of convex geometry. Equipped with the tools of modern geometric combinatorics, we re-examine the structure of these compactifications, leveraging these tools to understand the relationships between different compactifications and to compute invariants of the resulting schemes.

## Speaking at the seminar:

We are more than delighted to have people volunteer to speak at the seminar! Virtually any topic that falls within the scope of combinatorics, algebra, and/or geometry is welcome -- particularly those that cover all three.

## To volunteer to speak:

Please email either one of the organizers (see above) with the following information:

Talk title

Abstract

Preferred date

Please refer to dates marked "TBA" in schedule for the quarter.

(Optional) Anything else you wish for us to take into account!

# Autumn 2024 Talks

## Date TBA

### Title: Convex Algebraic Geometry of Higher-Rank Numerical Ranges

Speaker: Jonathan Niño-Cortes

Abstract: From the theory of quantum error correction, a generalization of numerical ranges called Higher-Rank Numerical Ranges emerged. It has been established that these sets are compact and convex subsets of the plane of complex numbers, giving a generalization of the famous Toeplitz-Haussdorff theorem. We aim to understand these sets further by applying the techniques of real algebraic geometry, projective duality and symbolic computation. Our goals include understanding the algebraic boundary of these sets and giving an algorithm that explicitly calculates higher-rank numerical ranges for a given complex square matrix.