Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in P n defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true for n≤3 and constructed counterexamples for every n≥4. Gordan and Noether and Franchetta gave classification of hypersurfaces in P 4 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.Obviously if the hypersurface X = V (f ) ⊂ P n is a cone (i.e. up to a linear change of coordinates f does not depend on all the variables), then the hessian polynomial

We prove a connectedness result for products of weighted projective spaces.

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In this paper we improve a classical result of Van de Ven which characterizes linear subspaces of as the only smooth closed subvarieties of for which the normal sequence splits (see [A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale (Bruxelles, 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]). Precisely we prove the following: Let X be a submanifold of of dimension ≥ 3, and Y ⊆ X a submanifold of X of dimension ≥ 2. Assume that Span(Y) = Span(X), where Span(X) is the smallest linear subspace of containing X. Then the exact sequence: splits if and only if X is a linear subspace of .