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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 is known. In this paper, we give an explicit (and simple) algorithm for computing a generic genus 4 curve from its theta hyperplanes.

Effective Reconstruction of Generic Genus 4 Curves from Their Theta Hyperplanes