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arXiv · 1204.4819

Components of the Hilbert scheme of space curves on low-degree smooth surfaces

Jan O. Kleppe·John C. Ottem
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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.

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Jan O. Kleppe, John C. Ottem. 2015-01-16. Components of the Hilbert scheme of space curves on low-degree smooth surfaces. https://arxiv.org/abs/1204.4819

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