Brill-Noether theory of Hilbert schemes of points on surfaces

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 descriptionof 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.