The Hilbert Schemes of Degree Three Curves are Connected

1996/03/14 by Scott Nollet, Nollet, Scott · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9603011

20 pages, Latex

arxiv created 1996/03/14 · arxiv updated 2009/11/30

Abstract

In this paper we show that the Hilbert scheme H(3,g) of locally Cohen-Macaulay curves in \Pthree of degree three and genus g is connected. In contrast to H(2,g), which is irreducible, H(3,g) generally has many irreducible components (roughly -g/3 of them). To show connectedness, we classify the curves (giving particular attention to the triple lines), determine the irreducible components, and give flat families over \Aone to show that the components meet. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.

Cited by

• Bourbaki degree of pairs of projective surfaces
• Codimension one distributions of degree 3 on the three-dimensional projective space
• Quartic curves in the quintic del Pezzo threefold

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