Reconstruction of hypersurfaces from their invariants
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913873279320064 |
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| author | Bouchet, Thomas |
| author_facet | Bouchet, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstruction of hypersurfaces from their invariants Bouchet, Thomas Commutative Algebra Algebraic Geometry 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. |
| title | Reconstruction of hypersurfaces from their invariants |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2403.17490 |