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The Hessian correspondence of hypersurfaces of degree 3 and 4

  • Javier Sendra
Raum P801 Universität Leipzig (Leipzig)

Abstract

Let $\mathcal{V}$ be a hypersurface in a $n$--dimensional projective space The Hessian map is a rational map from $\mathcal{V}$ to the projective space of symmetric matrices that sends $p\in \mathcal{V}$ to the Hessian matrix of the defining polynomial of $\mathcal{V}$ specialized at $p$. The Hessian correspondence is the map that sends a hypersurface to Zariski closure of its image through the Hessian map. In this paper, we study this correspondence for the cases of hypersurfaces of degree $3$ and $4$. 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. Moreover, we provide effective algorithms for recovering a hypersurface from its image through the Hessian map for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even.

seminar
12/8/22 1/25/24

Seminar on Algebra and Combinatorics

Universität Leipzig Seminargebäude 213

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail