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arXiv · 1801.01101

The Hilbert scheme of space curves sitting on a smooth surface containing a line

Jan O. Kleppe
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Abstract

We continue the study of maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s > 3, we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of H(d,g)_{sc}. In another range, close to the boarder of the nef cone, we describe for s=4 and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for s > 5 (resp. s=5), thus making progress to recent results of Kleppe and Ottem in [28]. For s=3 we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range.

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Jan O. Kleppe. 2018-01-03. The Hilbert scheme of space curves sitting on a smooth surface containing a line. https://doi.org/10.1007/s12215-016-0266-6

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