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Reconstruction of curves from their theta hyperplanes in genera $6$ and $7$

Türkü Özlüm Çelik, David Lehavi

TL;DR

The paper addresses reconstructing a generic genus $6$ or $7$ canonical curve from its theta hyperplanes, delivering a practical inverse Torelli map and a complete Schottky-locus description for these genera. It combines algebraic constructions with certified numerical methods, specializing to Wiman's sextic and Fricke–Macbeath curves as test cases, to certify theta hyperplanes, Steiner sets, and the key dimension data. The main contribution is an explicit reconstruction formula tied to the kernel of $\pi^2_{K_C}$ and the intersections of the $V_{2,\alpha}$ spaces, justified through a carefully designed numerical certification pipeline. This yields an effective, algebraic handle on the Schottky problem in genus $6$ and $7$ and provides a concrete computational framework for high-genus curve reconstruction via theta data, with potential extensions to genus $8$ and beyond.

Abstract

We derive a formula for reconstructing a generic complex canonical curve $C$ of genus 6 and 7 in terms of the theta hyperplanes of $C$. Hence, we get a generic inverse to the Torelli map, as well as a complete description of the Schottky locus in these genera. The computational part of the proof relies on a certified numerical argument.

Reconstruction of curves from their theta hyperplanes in genera $6$ and $7$

TL;DR

The paper addresses reconstructing a generic genus or canonical curve from its theta hyperplanes, delivering a practical inverse Torelli map and a complete Schottky-locus description for these genera. It combines algebraic constructions with certified numerical methods, specializing to Wiman's sextic and Fricke–Macbeath curves as test cases, to certify theta hyperplanes, Steiner sets, and the key dimension data. The main contribution is an explicit reconstruction formula tied to the kernel of and the intersections of the spaces, justified through a carefully designed numerical certification pipeline. This yields an effective, algebraic handle on the Schottky problem in genus and and provides a concrete computational framework for high-genus curve reconstruction via theta data, with potential extensions to genus and beyond.

Abstract

We derive a formula for reconstructing a generic complex canonical curve of genus 6 and 7 in terms of the theta hyperplanes of . Hence, we get a generic inverse to the Torelli map, as well as a complete description of the Schottky locus in these genera. The computational part of the proof relies on a certified numerical argument.
Paper Structure (14 sections, 29 theorems, 76 equations, 1 figure)

This paper contains 14 sections, 29 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations, statement of the result, and outline of the proof
  3. Numerically certifying a single theta hyperplane
  4. Numerically certifying Steiner sets given certified theta hyperplanes
  5. Numerically certifying the dimensions of $V_{2,\alpha}$ and their intersection
  6. A small matter of programming: Concluding the proof of the main theorem
  7. Generating non-certified theta hyperplanes
  8. Errors inherent to using SVD
  9. The numerics of \ref{['sec:certificationThetaHyperplanes']}
  10. Errors incurred by certifying the Steiner set in \ref{['sec:certificationSteiner']}
  11. Errors incurred by certifying \ref{['sec:dimcert']}, and the proof of the main theorem
  12. Testing
  13. Computing theta hyperplanes numerically
  14. Some results in computational linear algebra

Key Result

Lemma 2.3

The codimension of $V_{n,\alpha}$ is at least $d_\alpha$.

Figures (1)

  • Figure 1: An illustration of the conic approximation argument for a plane quartic. Note that both $P,v_x$ and the tangency point are on the quartic curve, and that the conic approximation is at $P$ -- not at the tangency point.

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4: $n = 1$
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • Lemma 2.8
  • ...and 63 more