The Hilbert schemes of locally Cohen–Macaulay curves in $${\mathbb{P}^3}$$ may after all be connected

Abstract

Progress on the problem whether the Hilbert schemes of locally Cohen–Macaulay curves in \({\mathbb{P}^3}\) are connected has been hampered by the lack of an answer to a question raised by Robin Hartshorne in (Commun. Algebra 28:6059–6077, 2000) and more recently in (American Institute of Mathematics, Workshop components of Hilbert schemes, problem list, 2010. http://aimpl.org/hilbertschemes): does there exist a flat irreducible family of curves whose general member is a union of d ≥ 4 disjoint lines on a smooth quadric surface and whose special member is a locally Cohen–Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d we construct a family with the required properties, whose special fiber is an extremal curve in the sense by Martin-Deschamps and Perrin (Ann. Sci. E.N.S. 4e Série 29:757–785, 1996). From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.