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 nonempty 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.