Brill–Noether Theory of Hilbert Schemes of Points on Surfaces

Abstract

Abstract We show that Brill–Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$, where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino’s description [9] of the Hilbert–Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill–Noether loci given by nested Hilbert schemes.