Tropicalization of classical moduli spaces

TL;DR

This paper explores the tropicalization of classical moduli spaces related to complex reflection arrangements, providing new combinatorial insights into their structure, especially focusing on the Burkhardt quartic and genus 2 curves.

Contribution

It introduces a method to tropicalize classical moduli spaces using matroid theory and applies it to several key examples, including the Burkhardt quartic.

Findings

Tropicalization of the Burkhardt quartic results in a 3D fan in 39D space

Provides a combinatorial approach to tropical moduli of genus 2 curves

Connects concrete geometric models with abstract tropical techniques

Abstract

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.