Details:
Title  The Hilbert schemes of locally CohenMacaulay curves in $mathbbP^3$ may after all be connected.  Author(s)  Paolo Lella, Enrico Schlesinger  Type  Article in Journal  Abstract  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 MartinDeschamps and Perrin (Ann. Sci. E.N.S. 4 e 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.  Keywords  Hilbert scheme, Locally Cohen–Macaulay curve, Initial ideal, Weight vector, Groebner bases  ISSN  00100757; 20384815/e 
URL 
http://link.springer.com/article/10.1007%2Fs1334801200623 
Language  English  Journal  Collect. Math.  Volume  64  Number  3  Pages  363372  Publisher  Springer, Milan; Universitat de Barcelona, Institut de Matem`atica, Barcelona  Year  2013  Edition  0  Translation 
No  Refereed 
No 
