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

2012/04/21 by Jan O. Kleppe, Kleppe, Jan O., John Christian Ottem +1 · 1 citation
Arts and Humanities · Mathematics · #14C05 (Primary) #14C20 #14H50 (Secondary) #14J28 #14K30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #North African History and Literature

paper · pdf · doi:10.48550/arxiv.1204.4819

openalex publication_date 2012/04/21 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We study maximal families W of the Hilbert scheme, H(d,g)_sc, of smooth\nconnected space curves whose general curve C lies on a smooth surface S of\ndegree s. We give conditions on C under which W is a generically smooth\ncomponent of H(d,g)_sc and we determine dim W. If s=4 and W is an irreducible\ncomponent of H(d,g)_sc, then the Picard number of S is at most 2 and we\nexplicitly describe, also for s > 4, non-reduced and generically smooth\ncomponents in the case Pic(S) is generated by the classes of a line and a\nsmooth plane curve of degree s-1. For curves on smooth cubic surfaces the first\nauthor finds new classes of non-reduced components of H(d,g)_sc, thus making\nprogress in proving a conjecture for such families.\n

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