On the birational geometry of Hilbert schemes of points and Severi divisors

Abstract

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of and when there is a birational morphism between surfaces. In this setting, embeds in and we ask if the restriction of the stable base locus decomposition of yields the respective decomposition in if the weak Lefschetz Principle holds. Even though the stable base loci in fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of to is equal to the stable base locus decomposition of We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that X is the projective plane, these divisors yield walls of the SBLD for some cases.