On the Gorenstein locus of some punctual Hilbert schemes

AbstractLet k be an algebraically closed field and let HilbdG(PkN) be the open locus of the Hilbert scheme Hilbd(PkN) corresponding to Gorenstein subschemes. We prove that HilbdG(PkN) is irreducible for d≤9. Moreover we also give a complete picture of its singular locus in the same range d≤9. Such a description of the singularities gives some evidence to a conjecture on the nature of the singular points in HilbdG(PkN) that we state at the end of the paper