Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur

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a b s t r a c tIt is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E d 1 ,...,dn on P N defined as the kernel of a general epimorphismIn this note we restrict our attention to the case of syzygy bundles E d,n on P N associated to n generic forms+N−2 and (N, n, d) = (2, 5, 2). This bound improves, in general, the bound n ≤ d(N + 1) given by Hein (2008 [2]), Appendix A.In the last part of the paper, we study moduli spaces of stable rank n − 1 vector bundles on P N containing syzygy bundles. We prove that if N + 1 ≤ n ≤ d+2 2 + N − 2, N = 3 and (N, n, d) = (2, 5, 2), then the syzygy bundle E d,n is unobstructed and it belongs to a generically smooth irreducible component of dimension n d+N N − n 2 , if N ≥ 4, and n d+2 2 + n d−1 2 − n 2 , if N = 2.