Vol. 6, No. 4, 2012

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Detaching embedded points

Dawei Chen and Scott Nollet

Vol. 6 (2012), No. 4, 731–756

Suppose that closed subschemes X Y N differ at finitely many points: when is Y a flat specialization of X union isolated points? Our main result says that this holds if X is a local complete intersection of codimension two and the multiplicity of each embedded point of Y is at most three. We show by example that no hypothesis can be weakened: the conclusion fails for embedded points of multiplicity greater than three, for local complete intersections X of codimension greater than two, and for nonlocal complete intersections of codimension two. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus and show the smoothness of the Hilbert component whose general member is a plane curve union a point in 3.

Hilbert schemes, embedded points
Mathematical Subject Classification 2010
Primary: 14B07
Secondary: 14H10, 14H50
Received: 17 December 2010
Revised: 10 May 2011
Accepted: 30 June 2011
Published: 25 July 2012
Dawei Chen
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
851 S. Morgan St.
Chicago, IL 60607
United States
Department of Mathematics
Boston College
Chestnut Hill, MA 02467
United States
Scott Nollet
Department of Mathematics
Texas Christian University
Box 298900
Fort Worth, TX 76129
United States