Components of the Hilbert scheme of space curves on low-degree smooth surfaces
arXiv:1204.4819
Abstract
We study maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d,g)_{sc} and we determine dim W. If s=4 and W is an irreducible component of H(d,g)_{sc}, then the Picard number of S is at most 2 and we explicitly describe, also for s > 4, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s-1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d,g)_{sc}, thus making progress in proving a conjecture for such families.
In this version, with J.C. Ottem as full coauthor, we extend previous results for curves on smooth cubic and quartic surfaces (both the unobstructedness criterion and the study of certain components of the Hilbert scheme) to curves on smooth surfaces of any degree > 2. The appendix on curves on cubic surfaces are due to the first author. To appear in International Journal of Mathematics. 23 pages