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The Hessian correspondence of hypersurfaces of degree 3 and 4

Javier Sendra-Arranz

TL;DR

The paper introduces and analyzes the Hessian correspondence $H_{d,n}$ and its Waring-rank restricted version $H_{d,n,k}$ for degree $d$ hypersurfaces in ${\bf P}^n$, with a focus on $d=3$ and $d=4$ and $k\le n+1$. It proves that for these ranges, $H_{d,n,k}$ is generically finite with degree $2^{k-1}$ when $d$ is odd and birational when $d$ is even, and it shows $H_{3,n}$ is birational for $n\ge 2$ while $H_{4,n}$ is birational for all $n$. The work introduces the variety of $k$-gradients $\phi_k$ and establishes their irreducible components as $F(\mathcal Z_l,k)$, along with birationality results for the associated maps $\alpha_k$. It also provides effective reconstruction algorithms for recovering the original hypersurface from its Hessian in the cases $(d,n)=(3,{\ge}1)$ and $(d,n)=(4,{\rm even})$, and it develops a detailed geometric framework around Veronese embeddings and first-order syzygies to underpin these results. The findings advance understanding of the Hessian geometry of low-degree hypersurfaces and offer practical tools for inverse problems in this setting, with potential connections to the classical Hesse problem.

Abstract

Let $X$ be a hypersurface, of degree $d$, in an $n$--dimensional projective space. The Hessian map is a rational map from $X$ to the projective space of symmetric matrices that sends a point $p\in X$ to the Hessian matrix of the defining polynomial of $X$ evaluated at $p$. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. In this paper, we study this correspondence for hypersurfaces with Waring rank at most $n+1$ and for hypersurfaces of degree $3$ and $4$. We prove that, for hypersurfaces with Waring rank $k\leq n+1$, the map is birational onto its image for $d$ even, and it is generically finite of degree $2^{k-1}$ for $d$ odd. We prove that, for degree $3$ and $n=1$, the map is two to one, and that, for degree $3$ and $n\geq 2$, and for degree $4$, the Hessian correspondence is birational. In this study, we introduce the $k$--gradients varieties and analyze their main properties. We provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even.

The Hessian correspondence of hypersurfaces of degree 3 and 4

TL;DR

The paper introduces and analyzes the Hessian correspondence and its Waring-rank restricted version for degree hypersurfaces in , with a focus on and and . It proves that for these ranges, is generically finite with degree when is odd and birational when is even, and it shows is birational for while is birational for all . The work introduces the variety of -gradients and establishes their irreducible components as , along with birationality results for the associated maps . It also provides effective reconstruction algorithms for recovering the original hypersurface from its Hessian in the cases and , and it develops a detailed geometric framework around Veronese embeddings and first-order syzygies to underpin these results. The findings advance understanding of the Hessian geometry of low-degree hypersurfaces and offer practical tools for inverse problems in this setting, with potential connections to the classical Hesse problem.

Abstract

Let be a hypersurface, of degree , in an --dimensional projective space. The Hessian map is a rational map from to the projective space of symmetric matrices that sends a point to the Hessian matrix of the defining polynomial of evaluated at . The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. In this paper, we study this correspondence for hypersurfaces with Waring rank at most and for hypersurfaces of degree and . We prove that, for hypersurfaces with Waring rank , the map is birational onto its image for even, and it is generically finite of degree for odd. We prove that, for degree and , the map is two to one, and that, for degree and , and for degree , the Hessian correspondence is birational. In this study, we introduce the --gradients varieties and analyze their main properties. We provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree and , and for degree and even.
Paper Structure (13 sections, 35 theorems, 90 equations, 2 algorithms)

This paper contains 13 sections, 35 theorems, 90 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hessian correspondence and Waring rank
  3. Hessian correspondence of degree $3$
  4. Case $n=1$
  5. Variety of $k$--gradients
  6. Birationality of $\alpha_k$
  7. Case $n\geq 2$
  8. Irreducible components of the variety of $k$--gradients
  9. Hessian correspondence of degree $4$
  10. Case $n=1$
  11. Case $n\geq 2$
  12. Reconstruction algorithm for $n$ even
  13. Conclusions

Key Result

Theorem 1

For $d\geq 3$ and $k\leq n+1$, $H_{d,n,k}$ is generically finite. For $d$ odd, it has degree $2^{k-1}$. For $d$ even, $H_{d,n,k}$ is birational.

Theorems & Definitions (77)

  • Theorem : Theorem \ref{['theo:sym hessian']}
  • Theorem : Theorem \ref{['theo:inj H3 1']}
  • Theorem : Theorem \ref{['theo:4,n']}
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 67 more