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arXiv · 2403.17490

Reconstruction of hypersurfaces from their invariants

Thomas Bouchet
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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.

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Thomas Bouchet. 2025-06-04. Reconstruction of hypersurfaces from their invariants. https://arxiv.org/abs/2403.17490

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