Reconstruction of hypersurfaces from their invariants

TL;DR

This paper introduces an explicit algorithm to reconstruct generic homogeneous polynomials from their invariants, enabling the recovery of algebraic curves of genus 3 and 4 from their invariants with coefficients in their field of moduli.

Contribution

It provides a novel, explicit method using covariants and Veronese embeddings to determine the polynomial's linear equivalence class from invariants.

Findings

Reconstruction formulas for genus 4 non-hyperelliptic curves.

Reconstruction formulas for genus 3 non-hyperelliptic curves.

Coefficients lie in the field of moduli.

Abstract

Let $K$ be a field of characteristic $0$. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial $f$ under the linear action of $\mathrm{GL}_n$ or $\mathrm{SL}_n$, returns a polynomial differing from $f$ only by a linear change of variables with coefficients in a finite extension of $K$. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. In both cases, the coefficients of the reconstructed curve lie in its field of moduli.