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

2008/02/07 by Alice Garbagnati, Garbagnati, Alice, Flavia Repetto +1 · 2 citations
Mathematics · #14J70 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.0802.0959

openalex publication_date 2008/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Hesse claimed that an irreducible projective hypersurface in \PPⁿ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for n≤ 3 and constructed counterexamples for every n≥ 4. Gordan and Noether and Franchetta gave classification of hypersurfaces in \PP⁴ 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.

Cited by

• Hesse's Redemption: Efficient Convex Polynomial Programming
• The Hesse Pencil Variety

Related

• On cubic hypersurfaces with vanishing hessian
• On higher Hessians and the Lefschetz properties
• On the theory of Gordan-Noether on homogeneous forms with zero Hessian (Improved version)
• On the Lefschetz locus in Gor(1,n,n,1)
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