  Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay
curves in projective 3 space are connected has been hampered by the lack of an
answer to a question that was raised by Robin Hartshorne in his paper "On the
connectedness of the Hilbert scheme of curves in projective 3 space" Comm.
Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM
workshop Components of Hilbert Schemes available at
http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of
curves whose general member is a union of d 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 of Martin-Deschamps and Perrin. 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.
