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

摘要

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 st ble divisorsbase locus decomposition (SBLD) of the cones of effective of and when there is a birational morphism f : X Y between surfaces. In this setting, Ni fyI"Ij embeds in N' (x["1), and we ask if the restriction of the stable base locus decomposition of Ni (XInI) yields the respective decomposition in N1 (YInl) i.e., if the weak Lefschetz Principle holds. Even though the stable base loci in Ni(XI'll) fails to provide informaion about how the two decompositions interact, we show that the restricion of the augmented stable base loci of to is equal to the stable base locus decomposition of YInI. We also exhibit effective divisors induced by Seven 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.