Schedule for: 23w5101 - Curves: Algebraic, Tropical, and Logarithmic

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Gather for informal Meet and Greet at BIRS Lounge in Professional Development Centre (2nd floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This talk will be an introduction to tautological intersection theory on the moduli space of curves. This studies a part of the intersection ring which has a rich combinatorial structure and yet contains many meaningful geometric cycles, (e.g. Hurwitz loci, GW-cycles etc). Many of these cycles have the undesirable property of not intersecting the boundary of the moduli space of curves in a dimensionally transverse way. Recently developed tool of logarithmic geometry allow to remedy this situation, at the cost of modifying the boundary structure of the moduli space of curves. We will survey some recent results that employ these techniques, and discuss in detail the clarifying example of Hurwitz cycles.

Brill--Noether theory provides the bridge between the classical study of curves embedded in projective space, and the more modern notion of an abstract curve. In this talk I'll introduce the basic questions and results in Brill--Noether theory, as well as some of the degenerative techniques used to prove these results.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will provide an overview of the concepts and methods of linear series on graphs and tropical curves, with an emphasis on enumeration and parameterization. The guiding example will be an elegant bijection found by Cools, Draisma, Payne, and Robeva, between standard Young tableaux on a rectangular partition and certain linear series on a chain of loops. From here, I will explain some generalizations and recent directions, especially the metamorphosis of this bijection to more recent work on Hurwitz--Brill--Noether theory, emphasizing the role of the affine symmetric group.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk I will describe a new definition, joint with Bivas Khan, for a tropical vector bundle on a subvariety of a tropical toric variety. This builds on the tropicalizations of toric vector bundles. I will discuss when these bundles do and do not behave as in the classical setting, and examples of vector bundles on curves.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we determine when there is a Brill--Noether curve of given degree and given genus that passes through a given number of general points in any projective space.

I will describe an extension to nodal curves of the space of linear series on smooth curves. This is work in progress with Luca Battistella and Francesca Carocci.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe moduli spaces of rational stable curves with finite cyclic action. The intersection theory of these moduli spaces is governed by the combinatorics of multimatroids. These multimatroids, introduced by Bouchet, generalize matroids, delta-matroids, and matroidal structures appearing in ribbon graphs. The geometry of these moduli spaces in turn inform us about statements of multimatroids. Based on past and present joint works with Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Rohini Ramadas.

I will discuss recent work, joint with Stefano Serpente and Claudia Yun, on the permutation group representations determined by the rational cohomology of moduli spaces of weighted stable curves with heavy/light weight data. Time permitting, I will discuss applications of our formula to moduli spaces of quasimaps to projective space, in ongoing joint work with Terry Song.

Logarithmic and orbifold structures provide two independent ways to model curves in a variety tangent to a divisor. Simple examples demonstrate that the moduli spaces and associated enumerative invariants differ, but a more structural explanation of this defect has remained elusive. We identify "birational invariance" as the key property distinguishing the two theories. The logarithmic theory is stable under blowups of the target variety, while the orbifold theory is not. By identifying a suitable system of blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. The correct system of blowups is describes in terms of tropical curves. No prior knowledge of Gromov-Witten theory will be assumed. This is joint work with Luca Battistella and Dhruv Ranganathan.

I will discuss how novel ideas from non-abelian Brill-Noether theory and from the theory of resonance varieties can be used to prove that the moduli space of Prym varieties of genus 13 is of general type (and that the moduli space of curves of genus 16 is uniruled). For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will briefly indicate the use of tropical geometry in order to establish an essential case of the Strong Maximal Rank Conjecture, necessary to carry out this program. Joint work with Jensen and Payne (respectively Verra).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

A basic question is understanding how the Hilbert/ Quot scheme of a projective variety X changes when we degenerate X. The key to answering this question is to study the geometry of a pair (X,D) with D a divisor on X. I will discuss compact moduli spaces called the logarithmic Hilbert/ Quot schemes which track this transverse geometry. The tropical version of a Hilbert scheme plays an important role in this story.

A hyperplane section of a canonically embedded, non-hypereliptic smooth curve of genus g consists of 2g-2 points in g-2 dimensional projective space. From work by Dolgachev and Ortland it is known that point configurations obtained in such a way are self-associated. We interpret this notion in terms of matroids: A generic hyperplane section of a canonical curve gives rise to an identically self-dual matroid. This can also be seen as a combinatorial shadow of the Riemann-Roch theorem. We compute these sets of matroids up to rank 5. Building on works by Bath, Mukai and Petrakiev, we investigate the question, which self-dual point configurations can be obtained by a hyperplane section of a canonical curve. Self-dual point configurations are parametrized by a subvariety of the Grassmannian Gr(n,2n) and its tropicalization. This is well understood for n=3, while n=4 provides more challenges. This project is joined work with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In recent work, Chan--Galatius--Payne have developed tools to study the topology of tropical moduli spaces. The original motivation was the application to the moduli space of (marked) curves, in which case the tropical computations gave new insight into certain graded pieces of the cohomology of the algebraic moduli space $\mathcal{M}_{g,n}$. In our current project, we use these tools and apply them to the moduli space of cyclic unramified tropical covers with Galois group $\mathbb{Z}/p\mathbb{Z}$. These spaces are similar in behavior but mildly more complicated than $M_{g,n}^{trop}$. I will present some general results on certain contractible loci, which is a crucial simplification step. Moreover, I will also present some explicit computations which show that the homotopy type of the moduli space in genus $g = 2$ is a wedge of spheres. This project is motivated by work in progress by Pedro Souza, which promises application to the cohomology of algebraic moduli spaces. This is joint work in progress with Yassine El Maazouz, Paul Alexander Helminck, Pedro Souza, and Claudia Yun.

In recent years, it has been realized that the intersection theory of the moduli space of curves can be refined if one takes into account its logarithmic structure: one is led to an intersection theory that mixes algebraic and tropical cycles together. In this talk, I will survey some of the techniques, ideas, and results of this approach, through the examples of the double ramification and Brill-Noether cycles.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

p-adic heights have been a rich source of explicit p-adic analytic functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights for line bundles on abelian varieties from p-adic adelic metrics. This uses p-adic Arakelov theory developed by Besser, and is a non-Archimedean analogue of Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for finding rational points. Time permitting, we will also mention how a certain harmonicity formula for local height functions plays a key role in the explicit computation of some necessary p-adic constants. This is joint work with Amnon Besser and Steffen Mueller.

I will describe some connections between arithmetic/Arakelov geometry and non-archimedean/tropical geometry. The interplay arises from the study of analytic invariants on degenerating families of curves and abelian varieties, as well as the theory of heights of abelian varieties. (Based on recent and ongoing projects with Robin de Jong and Robert Wilms.)

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (over the moduli space of curves) Brill-Noether classes. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. This is joint with Nicola Pagani.

The famous Baker-Norine-Riemann-Roch theorem for graphs led to an intensive study of the properties of tropical complete linear series with many beautiful applications. For further applications to the geometry of algebraic curves and their moduli, it is essential also to have a framework in which to study tropicalizations of incomplete linear series, such as images of multiplication maps. I will discuss steps toward the creation of such a theory of tropical linear series and applications of a preliminary version of this theory to the Kodaira dimensions of $M_{22}$, $M_{23}$, and $R_{13}$. Joint work with Farkas and Jensen.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Tropical universal Jacobians are moduli spaces parametrizing tropical curves along with a divisor on it. In the last few years, they have been studied from different perspectives and in different categories. In the talk, I will try to give an account for different incarnations of tropical universal Jacobians and illustrate how can these be realized as the tropicalization of the moduli space of universal compactified Jacobians. Part of the talk will be based on joint work with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

Glueing maps have long played a central role in the study of moduli spaces of curves. At the beginning of this week we have seen that blowing up the moduli space can give very useful extra flexibility. However, blowups and glueing maps do not interact nicely. We will illustrate the problem by concrete examples, describe a fix by making log curves more logarithmic, and present some applications and open questions. This is joint work with Pim Spelier.