On higher Hessians and the Lefschetz properties

2015/06/21 by Gondim, Rodrigo
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.06387

Abstract

We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given N≥ 3, d ≥ 3 and 2≤ k < (d)/(2) there are infinitely many irreducible hypersurfaces X = V(f)⊂ ℙ^N, of degree deg(f)=d, not cones and such that their Hessian determinant of order k, hess^k_f, vanishes identically. The vanishing of higher Hessians is closely related with the Strong (or Weak) Lefschetz property for standard graded Artinian Gorenstein algebra, as pointed out firstly in \citeWa1 and later in \citeMW. As an application we construct for each pair (N.d) ≠ (3,3),(3,4) infinitely many standard graded Artinian Gorenstein algebras A, of codimension N+1 ≥ 4 and with socle degree d ≥ 3 which do not satisfy the Strong Lefschetz property, failing at an arbitrary step k with 2≤ k

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