On higher Hessians and the Lefschetz properties

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 irreducible hypersurfaces X = V (f ) ⊂ P N , of degree deg(f ) = d, not cones and such that their Hessian 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 first in [Wa1] and later in [MW]. As an application we construct for each pair (N, d) = (3, 3), (3, 4), 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 < d 2 . We also prove that for each pair (N, d), N ≥ 3 and d ≥ 3 except (3, 3), (3, 4), (3, 6) and (4, 4) there are standard graded Artinian Gorenstein algebras of codimension N + 1, socle degree d, with unimodal Hilbert vectors and which do not satisfy the Weak Lefschetz property.

The Jordan type of graded Artinian Gorenstein algebras

“…Hence J A = 5 1 ⊕ 3 n−1 ⊕ 1 a−n . The second assertion, for n = 5, uses the same arguments of [Go,Thm. 3.5].…”

“…Proof. By Gordan-Noether Theorem (see [GN, Ru, Go]), an algebraic dependence among the partial derivatives of f implies that hess f = 0 (see [Go,Proposition 3.10]). Since ∂f ∂xi = g 1 ∈ K[u 1 , .…”

“…By Gordan-Noether Theorem, there is no hypersurfaces in P n−1 with n ≤ 4 not a cone and having vanishing Hessian (see [Go,Theorem 3.9]). The classification of cubics with vanishing Hessian in low dimension is part of the classical work of Perazzo (see [Pe]) this classical work was revisited in [GRu].…”

Complete intersection Jordan types in height two

“…There have been many studies of graded Artinian Gorenstein algebras satisfying the strong or weak Lefschetz property (see [7] and the references cited there). Recently, there have been studies of more general questions about the Jordan type of pairs ( , A) (see [4,5,7,10,16] and references cited.) By a result of F.H.S.…”

“…We denote by R = k[x, y] the polynomial ring in two variables over k. We will consider Artinian Gorenstein (so by F.H.S. Macaulay's result complete intersection) algebras A = R/ Ann F , where F ∈ E = k[X, Y ] is the Macaulay dual generator of A. T. Maeno and J. Watanabe in 2009 introduced a method of using higher Hessians to determine the strong or weak Lefschetz properties of a graded Artinian algebra [16]; this was further developed and used by T. Maeno and Y. Numata [15] and by R. Gondim and colleagues [4,5,3]. The Hilbert function of a graded CI quotient A of R satisfies H(A) = T , a symmetric sequence of the form T = (1 0 , 2 1 , .…”