2017 CMS Winter Meeting

Waterloo, December 8 - 11, 2017

Toric Geometry
Org: Matthew Satriano (University of Waterloo) and Greg Smith (Queen's Univeristy)

GRAHAM DENHAM, University of Western Ontario

LAURA ESCOBAR, University of Illinois at Urbana-Champaign

ANTONELLA GRASSI, University of Pennsylvania

NATHAN ILTEN, Simon Fraser University

KELLY JABBUSCH, University of Michigan-Dearborn

KALLE KARU, University of British Columbia

KIUMARS KAVEH, University of Pittsburgh

YOAV LEN, University of Waterloo

STEFFEN MARCUS, The College of New Jersey

JENNA RAJCHGOT, University of Saskatchewan

MARTIN ULIRSCH, University of Michigan, Ann Arbor
Tropical geometry of the Hodge bundle  [PDF]

The Hodge bundle is a vector bundle over the moduli space of smooth curves (of genus $g$) whose fiber over a smooth curve is the space of abelian differentials on this curve. We may define a tropical analogue of its projectivization as the moduli space of pairs $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and an effective divisor $D$ in the canonical linear system on $\Gamma$. This tropical Hodge bundle turns out to be of dimension $5g-5$, while it is a classical fact that the projective Hodge bundle has dimension $4g-4$. This means that not every pair $(\Gamma, D)$ in the tropical Hodge bundle arises as the tropicalization of a suitable element in the algebraic Hodge bundle.

In this talk I am going to outline a comprehensive (and completely combinatorial) solution to the realizability problem, which asks us to determine the locus of points in the tropical Hodge bundle that arise as tropicalizations. Our approach is based on recent work of Bainbridge-Chen-Gendron-Grushevsky-Möller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne.

This talk is based on joint work with Bo Lin, as well as on an ongoing project with Martin Möller and Annette Werner.

JAY YANG, University of Wisconsin-Madison