G. Casnati: Is the Gorenstein locus of the punctual Hilbert Scheme irreducible?

Abstract. Let H^G(n,d) be the Hilbert scheme parametrizing 0-dimensional Gorenstein subschemes of degree d of Pⁿ. A. Iarrobino stated that such a scheme is reducible if n≥6 and d≥14 asserting the existence of a scheme X in H^G(n,d) such that the tangent space to H^G(n,d) at the Hilbert point of X has low dimension. It is thus natural to ask if H^G(n,d) is irreducible in the complementary range. We prove that it is irreducible when d≤10 and we completely characterize its singularities when d≤9.