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

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Abstract

We continue the study of maximal families W of the Hilbert scheme, \( {{\mathrm{H}}}(d,g)_{sc}\), of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For \(s \ge 4\), we extend the two ranges where W is a unique irreducible (resp.  generically smooth) component of \( {{\mathrm{H}}}(d,g)_{sc}\). 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 \(s \ge 6\) (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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  1. Faculty of Technology, Art and Design, Oslo and Akershus University College of Applied Sciences, PO Box 4 ST, Olavs Plass, 0130, Oslo, Norway

    Jan O. Kleppe

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Kleppe, J.O. The Hilbert scheme of space curves sitting on a smooth surface containing a line. Rend. Circ. Mat. Palermo, II. Ser 66, 97–112 (2017). https://doi.org/10.1007/s12215-016-0266-6

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