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

Consider the Hilbert scheme of points on a higher dimensional affine space. Its component is elementary if it parameterizes irreducible subschemes. We characterize reduced elementary components in terms of tangent spaces and provide a computationally efficient way of finding such components. As an example, we find an infinite family of elementary and generically smooth components on the affine four-space. We analyse singularities and formulate a conjecture which would imply the non-reducedness of the Hilbert scheme. Our main tool is a generalization of the Bia lynicki-Birula decomposition for this singular scheme.

In this paper we introduce the open Waring rank of a form of degree d in n variables and prove the that this rank in bounded from above bywhenever n, d ≥ 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of [BBS] and giving, as far as we know, the best upper bound known.