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

Progress on the problem whether the Hilbert schemes of locally Cohen–Macaulay curves in P3 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