A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse

Alice Garbagnati; Flavia Repetto

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arXiv · 0802.0959

A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse

Alice Garbagnati·Flavia Repetto

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Abstract

Hesse claimed that an irreducible projective hypersurface in $\PP^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\leq 3$ and constructed counterexamples for every $n\geq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\PP^4$ with vanishing hessian and which are not cones. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.

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Alice Garbagnati, Flavia Repetto. 2008-02-07. A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse. https://arxiv.org/abs/0802.0959

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