Components of the Hilbert scheme of space curves on low-degree smooth surfaces
Journal article, Peer reviewed
Electronic version of an article published as kleppe, j. o., & ottem, j. c. (2015). components of the hilbert scheme of space curves on low-degree smooth surfaces. international journal of mathematics, 26(02), 1550017. © world scientific publishing company.
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OriginalversjonKleppe, J. O., & Ottem, J. C. (2015). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. International Journal of Mathematics, 26(02), 1550017. http://dx.doi.org/10.1142/S0129167X15500172
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 ≥ 5, 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.