The sum of the Betti numbers of smooth Hilbert schemes

Abstract

Recently, Skjelnes and Smith classified which Hilbert schemes on projective space are smooth in terms of integer partitions $$\lambda = (\lambda _1,\ldots ,\lambda _{r})$$ with $$r=0$$ , $$\lambda =(n+1)$$ , or $$n\geqslant \lambda _1\geqslant \cdots \geqslant \lambda _r \geqslant 1$$ . In particular, they found there to be seven families of smooth Hilbert schemes: one with $$r=0$$ or $$\lambda =(n+1)$$ , one with Hilbert schemes on the projective line or plane, 4 families with $$\lambda _r=1$$ , and one with $$\lambda _r\geqslant 2$$ . In this paper, we compute the sum of the Betti numbers for all of these families of smooth Hilbert schemes over projective space except the case $$\lambda _r\geqslant 2$$ .