Effective reconstruction of generic genus 4 curves from their theta hyperplanes

TL;DR

This paper presents a simple, explicit algorithm for reconstructing generic genus 4 curves from their theta hyperplanes, extending classical methods known for lower genera to higher genus cases.

Contribution

It introduces the first known explicit algorithm for reconstructing genus 4 curves from theta hyperplanes, filling a gap in algebraic geometry methods.

Findings

Provides an explicit reconstruction algorithm for genus 4 curves

Demonstrates the algorithm's effectiveness on generic cases

Extends classical genus 2 and 3 reconstruction techniques

Abstract

Effective reconstruction formulas of a curve from its theta hyperplanes are known classically in genus 2 (where the theta hyperplanes are Weierstrass points), and 3 (where, for a generic curve, the theta hyperplanes are bitangents to a plane quartic). However, for higher genera, no formula or algorithm are known. In this paper we give an explicit (and simple) algorithm for computing a generic genus 4 curve from it's theta hyperplanes.