Numerical reconstruction of curves from their Jacobians

TL;DR

This paper develops a computational method to reconstruct algebraic curves from their Jacobians using numerical algebraic geometry, verified through experiments up to genus 7.

Contribution

It introduces a practical numerical approach to solve the Torelli problem, extending previous theoretical work with effective algorithms.

Findings

Successfully reconstructed curves up to genus 7

Validated the method through numerical experiments

Demonstrated effectiveness of the approach in computational algebraic geometry

Abstract

We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7.