Irreducibility and singularities of some nested Hilbert schemes

Abstract

Let S be a smooth projective surface over C. We study the local and global geometry of the nested Hilbert scheme of points S[n,n+1,n+2]. In particular, we show that S[n,n+1,n+2] has klt singularities and compute its Picard group when h1(S,OS)=0. We also make some improvements on the codimension bounds of some incidence loci in S×S[n,n+1], which are used to establish irreducibility of S[n,n+1,n+2]. From the irreducibility of S[n,n+1,n+2], we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first effective example of a reducible nested Hilbert scheme, which allows us to show that S[n1,…,nk] is reducible for k>22.