Tropicalization of classical moduli spaces

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.

33 pages

References in corpus (1)

• Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

Cited by in corpus (6)

• Equations of tropical varieties
• Arithmetic aspects of the Burkhardt quartic threefold
• Tropical geometry of genus two curves
• Alternating trilinear forms on a 9-dimensional space and degenerations of (3,3)-polarized Abelian surfaces
• Detecting tropical defects of polynomial equations
• Lines on $p$-adic and real cubic surfaces