The Hilbert schemes of locally Cohen-Macaulay curves in may after all be connected

作者：Lella Paolo; Schlesinger Enrico^*

来源：Collectanea Mathematica, 2013, 64(3): 363-372.

DOI：10.1007/s13348-012-0062-3

• Hilbert scheme
• Locally Cohen-Macaulay curve
• Initial ideal
• Weight vector
• Groebner bases

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in 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 a parts per thousand yen 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. 4 (e) S,rie 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.

• 出版日期2013-9