Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in ℙⁿ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true forn≤3 and constructed counterexamples for everyn≥4. Gordan and Noether and Franchetta gave classification of hypersurfaces in ℙ⁴ with vanishing hessian and which are not cones, see [6, 5]. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.

中文翻译：

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戈登-诺瑟和弗朗切塔对黑森提出的问题的几何方法

黑森权利[7]（后来也在[8]），在ℙ不可约射影超曲面^Ñ通过与消失粗麻布决定簇的方程定义必然是一个锥体。戈尔丹和诺特证明在[6]，这是对真实Ñ ≤3和构造的反每Ñ ≥4。戈尔丹和诺特和Franchetta给超曲面的分类在ℙ ⁴具有消失粗麻布，哪些不是锥体，见[6,5]。在这里，我们用几何术语Gordan和Noether方法进行翻译，提供了这些结果的直接几何证明。