Schottky Algorithms: Classical meets Tropical

TL;DR

This paper introduces a novel approach combining classical and tropical geometry to address the Schottky problem, enabling the identification and computation of Jacobians and their embeddings in genus four.

Contribution

It provides new algorithms and implementations that connect classical algebraic geometry with tropical geometry for the Schottky problem in genus four.

Findings

Successful tropicalization of the Schottky-Igusa modular form

Algorithms for identifying Jacobians in genus four

Implementation of classical and tropical solutions

Abstract

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.