On the first infinitesimal neighborhood of a linear configuration of points in [formula omitted

Abstract

We consider the following open questions. Fix a Hilbert function h ̲ , that occurs for a reduced zero-dimensional subscheme of P 2 . Among all subschemes, X , with Hilbert function h ̲ , what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, Z , of X (i.e. the double point scheme supported on X )? Is there a minimum ( h ̲ min ) and maximum ( h ̲ max ) such function? The numerical information encoded in h ̲ translates to a type vector which allows us to find unions of points on lines, called linear configurations, with Hilbert function h ̲ . We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same. Even for those h ̲ for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on h ̲ ) that does occur. We prove the existence of h ̲ max , in general, and discuss h ̲ min . Our methods include liaison techniques.