Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyze the Gorenstein locus of the Hilbert scheme of d points on P^n i.e., the open subscheme parameterizing zero-dimensional Gorenstein subschemes of P^n of degree d. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when d <= 13 and find its components when d = 14. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by-product, we give equations of the fourth secant variety to the d-th Veronese reembedding of P^n for d >= 4.