On the irreducibility of Hessian loci of cubic hypersurfaces
arxiv(2024)
Abstract
We study the problem of the irreducibility of the Hessian variety
ℋ_f associated with a smooth cubic hypersurface V(f)⊂ℙ^n. We prove that when n≤5, ℋ_f is normal and
irreducible if and only if f is not of Thom-Sebastiani type, i.e., roughly,
one can not separate its variables. This also generalizes a result of Beniamino
Segre dealing with the case of cubic surfaces. The geometric approach is based
on the study of the singular locus of the Hessian variety and on infinitesimal
computations arising from a particular description of these singularities.
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