Schedule for: 16w5153 - Algebraic, Tropical, and Nonarchimedean Analytic Geometry of Moduli Spaces

A welcome drink will be served at the hotel.

Tropical geometry can be viewed as an efficient tool to organize degenerations. The techniques to count curves in surfaces via tropical geometry are related to the Fock space approach initiated by Cooper-Pandharipande, via floor diagrams (which can be viewed as the combinatorial essence of a tropical curve count) (following Block-Goettsche). Our own contribution relates the tropical and the Fock space approach for descendant Gromov-Witten invariants. (Joint work with Renzo Cavalieri, Paul Johnson and Dhruv Ranganathan.)

I will discuss a unified approach to tropicalizing moduli spaces of logarithmic stable maps to arbitrary logarithmically smooth targets using the geometry of Artin-Olsson fans. I will pay particular attention to superabundant case, exploring the relationship between obstructedness of logarithmic maps and the tropical realizability question. Some relatively transparent consequences of the framework include a new perspective on Speyer's well-spacedness condition, polyhedrality of the tropical realizability cycle, and a version of Murphy's Law for lifting problems for lifting tropical curves.

Artin fans are logarithmic algebraic stacks that are logarithmically \'etale over the base field. Despite their seemingly abstract definition, the geometry of Artin fans can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato’s notion of a fan. In this talk, following a rapid introduction to logarithmic geometry from a modular point of view, I am going to give an expository account of the theory of Artin fans and explain how Thuillier's non-Archimedean skeleton of a toroidal embedding can be understood as an analytification of the associated Artin fan. In the special case of a toric variety, this simply reduces to the fact that the Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient. Finally, Artin fans also provide the motivation for an ongoing joint project with R. Cavalieri, M. Chan, and J. Wise, in which we develop a stack-theoretic framework for the study of tropical moduli spaces.

I will discuss past and ongoing work with Melanie Wood on the (stabilization of) topology of various moduli spaces (and “discriminant loci” therein) by writing their classes in the Grothendieck ring in terms of motivic zeta functions (zeta functions in the Grothendieck ring).

In this talk we discuss the structure of tropicalisations of very affine surfaces obtained from del Pezzo surfaces of degree 5, 4, 3 by removing their \((-1)\)-curves. Classically these surfaces are the blow ups of \(\mathbf{P}^2\) in 4, 5, and 6 points (respectively) in general position. The \((-1)\)-curves on the surface then translate to an arrangement of metric trees. The main example will be del Pezzos of degree 3 which are cubic surfaces in \(\mathbf{P}^3\) and the \((-1)\) curves are exactly the 27 lines living on the surface. In this case there are only two non-degenerate combinatorial types of tropicalisations. These combinatorial models are found using three methods; tropicalisation of ideals, fibres of moduli maps and from point configurations in the tropical plane. This talk is based on joint work with Q. Ren and B. Sturmfels

Intersection theory is an extremely useful tool both in algebraic and tropical enumerative geometry. It can also be used to obtain algebraic/tropical correspondence theorems. The idea is to apply the tropicalization procedure to appropriate intersection products on suitable algebraic moduli spaces, thus obtaining the analogous tropical intersection products on the analogous tropical moduli spaces. For this to work in general, we need to be able to tropicalize cycle classes. I will present a suitable definition of tropical rational equivalence on extended cone complexes making this possible. As an application we obtain a correspondence theorem for genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.

I will classify the special divisors on a metric graph formed as a chain of loops with arbitrary edge lengths. In particular, I will state a necessary and sufficient criterion for a chain of loops to be Brill-Noether general (in the sense of Cools, Draisma, Payne, and Robeva). As an application, I will state an analog of the Brill-Noether theorem for algebraic curves of fixed gonality, by specializing to chains of loops with special edge lengths.

The Losev-Manin space parameterizing stable rational curves with the n+2 points weighted \(1^2, \epsilon^n\)​ is naturally associated with the root system of type \(A_{n-1}\)​. I will explain generalizations of this moduli space to the other root systems of ADE and extended ADE types. The talk is based on a joint work with Alan Thompson.

Classically, the exterior algebra provides an elegant description of the Plucker embedding.  In this talk I will present an analogue of this picture for matroids (and valuated matroids). Matroids can be thought of as objects defined over the boolean idempotent semiring B={0,1}.  We define a tropical analogue of exterior algebra for any quotient the free module B^n, and we show that a d-multivector w is a matroid if and only if the quotient of B^n that is dual to the kernel of wedging with w has d-th wedge power free of rank 1.

A metric graph is a graph together with an assignment of a positive real number to each edge. The isometry classes of metric graphs are points in a moduli space of metric graphs. Differences in the detailed definition leads to several flavors of moduli spaces of graphs with quite different homotopy types. Some flavors are related to automorphism groups of free groups, and some are related to moduli space of curves and tropical geometry. I will survey the different moduli spaces and the various maps between them, from a homotopy theoretic viewpoint. If time permits, I will discuss some new results joint with Melody Chan and Sam Payne.

We compactify moduli varieties of (un-embedded, polarized) varieties whose boundaries are the corresponding tropical moduli space (hence “cell complex”-like) i.e. the moduli of the corresponding class of tropical varieties. Cf., arXiv:1406.7772 

Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa demonstrates that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan and Max Hully on the geometry of analytic and tropical varieties over higher rank valued fields. I will explain how ideas coming from tame topology can be used to study the geometry of these spaces. I will explain the role that multi-stage degenerations play in this geometry, and if time permits, I will sketch an application of these ideas to the study of tropical and algebraic intersection multiplicities.

Motivated by mathematical physics, Göttsche and Shende have defined "quantized" versions of certain enumerative invariants for linear systems on surfaces. Block and Göttsche have proposed formulas for the corresponding multiplicities for tropical curves, refining the classical tropical multiplicities. In certain cases, they were able to prove a correspondence theorem for these refined invariants. In joint work with Sam Payne and Franziska Schroeter, we suggest a geometric interpretation of the Block-Göttsche multiplicities as \(\chi_y\) genera of semi-algebraic analytic domains over the field of Puiseux series. In order to define and compute these \(\chi_y\) genera, we use the theory of motivic integration developed by Hrushovski and Kazhdan and we explore its connections with tropical geometry.