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A geometrical approach to Gordan-Noether’s and Franchetta’s contributions to a question posed by Hesse. (English) Zbl 1180.14045

Classification of hypersurfaces of \(\mathbb P^n\) with vanishing Hessian is a classical still open problem. If \(n\leq 3\), the only such hypersurfaces are cones, but if \(n\geq 4\) this is no longer true, as shown by several authors including Gordan-Noether, Franchetta and Permutti. A detailed survey of this topic is given by C. Ciliberto, F. Russo and A. Simis [Adv. Math. 218, No. 6, 1759–1805 (2008; Zbl 1144.14009)]. The main result of the article under review is a precise geometric characterization of the hypersurfaces with vanishing Hessian that are not cones in the case of \(\mathbb P^4\).

MSC:

14J70 Hypersurfaces and algebraic geometry
14N05 Projective techniques in algebraic geometry
14N15 Classical problems, Schubert calculus

Citations:

Zbl 1144.14009
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References:

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