10:00am - 12:00pm
Chip-firing and tropical curves
The chip-firing game on metric graphs is a simple combinatorial model that serves as a tropical analogue of divisor theory on algebraic curves, and it has been an active and fruitful research direction over the last decade. The behaviors of chip-firing resemble, but not always completely match, the classical situation in algebraic geometry. So on one hand, chip-firing can often be used to prove results (old and new) in algebraic geometry; while on the other hand, the combinatorics of chip-firing is interesting and surprising in its own right. We will focus on three main topics: (I) Tropical analogues (or failure thereof) of classical results of algebraic curves, (II) applications of chip-firing in algebraic geometry and other subjects, and (III) complexity issues of computational problems related to chip-firing.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
The Tropical Ceresa Class
Let X be an algebraic curve of genus at least 2. The Ceresa cycle is the homologically trivial algebraic cycle X - [-1]_*X in Jac(X). It is trivial when X is hyperelliptic, and Ceresa proved that it is not algebraically equivalent to 0 for a generic curve of genus at least 3 defined over the complex numbers. We define the the tropical Ceresa class m(C) for a tropical curve C. This agrees with the l-adic Galois cohomology class associated to the Ceresa cycle of a curve defined over CC((t)) whose tropicalization is C. We show that m(C) is nontrivial whenever C has as a minor either K_4 or a loop of 3 loops. In particular, when C is 2-connected and unweighted, m(C) = 0 if and only if C is hyperelliptic. This is joint work with Jordan Ellenberg and Wanlin Li.
Tropical Prym Varieties
My talk revolves around combinatorial aspects of Prym varieties with applications to Brill-Noether theory. Prym varieties are a class of Abelian varieties that appear in the presence of covers between Riemann surfaces, and have deep connections with 2-torsion points of Jacobians, bi-tangent lines, and spin structures on curves. In my talk, I will describe the tropical version of Prym varieties in terms of chip-ring, and discuss the relation with their algebraic counterpart. As a consequence of the tropical construction we obtain new results in the geometry of special algebraic curves. This is joint work with Martin Ulirsch.
Equidistribution of tropical Weierstrass points
The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on abstract tropical curves, and explain how their distribution can be described in terms of electrical networks.
Submodular functions in tropical geometry: the existence of semibreak divisors
In the talk, I would like to show a situation where a statement in tropical geometry is proved using the submodular technique of combinatorial optimization.
Break divisors are a very useful concept in tropical geometry that were introduced by Mikhalkin and Zharkov. They give a system of representatives of divisor classes of degree g (where g is the genus). We introduce semibreak divisors, generalizing break divisors for degree less than the genus. We show that every effective divisor class of degree between 0 and g contains a semibreak representative. Semibreak divisors can be used to give elementary proofs for some properties of effective loci in tropical curves. Formerly, the only known proofs for these properties used the counterparts of these properties for algebraic curves.
To prove the existence of semibreak divisors in effective divisor classes of degree between 0 and g, we give a characterization of break divisors using a submodular function. Though the submodular function in our case is defined on an infinite set system (on those closed subsets of the metric graph that have finitely many path connected components), the problem has a quasi-discrete nature, which enables us to obtain a proof for the existence of semibreak divisors that very much resembles discrete arguments. We also obtain an algorithm that computes a semibreak representative for a given effective divisor, using a submodular minimization algorithm as subroutine. Joint work with Andreas Gross and Farbod Shokrieh.