Vol. 3, No. 4, 2009

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Ideals generated by submaximal minors

Jan O. Kleppe and Rosa M. Miró-Roig

Vol. 3 (2009), No. 4, 367–392

The goal of this paper is to study irreducible families Wt,tt1(b¯;a¯) of codimension 4, arithmetically Gorenstein schemes X n defined by the submaximal minors of a t × t homogeneous matrix A whose entries are homogeneous forms of degree aj bi. Under some numerical assumption on aj and bi, we prove that the closure of Wt,tt1(b¯;a¯) is an irreducible component of Hilbp(x)(n), show that Hilbp(x)(n) is generically smooth along Wt,tt1(b¯;a¯), and compute the dimension of Wt,tt1(b¯;a¯) in terms of aj and bi. To achieve these results we first prove that X is determined by a regular section of Y Y 2(s) where s = deg(detA) and Y n is a codimension-2, arithmetically Cohen–Macaulay scheme defined by the maximal minors of the matrix obtained deleting a suitable row of A.

Hilbert scheme, arithmetically Gorenstein, determinantal schemes
Mathematical Subject Classification 2000
Primary: 14M12, 14C05, 14H10, 14J10
Secondary: 14N05
Received: 3 October 2007
Revised: 12 December 2008
Accepted: 12 December 2008
Published: 15 June 2009
Jan O. Kleppe
Oslo University College
Faculty of Engineering
Postboks 4, St. Olavs Plass
Oslo, N-0130
Rosa M. Miró-Roig
University of Barcelona
Algebra and Geometry
Gran Via de les Corts Catalanes 585
Barcelona, 08007