A Torelli-like Theorem for Smooth Plane Curves

TL;DR

This paper explores a Torelli-like theorem for smooth plane curves, demonstrating that their period matrices can be significantly compressed by focusing on only four columns, which has implications for understanding their geometric and algebraic properties.

Contribution

It establishes a Torelli-like theorem showing that the period matrix of smooth plane curves can be reconstructed from just four of its columns, revealing a new compression property.

Findings

Period matrix of smooth plane curves is highly compressible.

Four columns of the period matrix suffice for reconstruction.

Implications for the Information-Theoretic Schottky Problem.

Abstract

The Information-Theoretic Schottky Problem treats the period matrix of a compact Riemann Surface as a compressible signal. In this case, the period matrix of a smooth plane curve is characterized by only 4 of its columns, a significant compression.