Tropical invariants for binary quintics and reduction types of Picard curves

TL;DR

This paper connects tropical invariants of binary quintics to the reduction types of Picard curves, providing a general framework for tropical invariants linked to group actions on varieties and their relation to moduli spaces.

Contribution

It introduces a novel framework for tropical invariants associated with group actions on varieties, specifically relating binary quintics to Picard curve reductions.

Findings

Reduction types of Picard curves expressed via tropical invariants.

Framework for tropical invariants linked to group actions on varieties.

Mapping binary forms to symmetrized Deligne-Mumford compactifications.

Abstract

We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. We also give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne-Mumford compactification $\overline{M}_{0,n}$.