The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after   all be connected
  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. 
Paolo Lella and Enrico Schlesinger
math.AG math.AC