Algebra & Number Theory

Ideals generated by submaximal minors

Jan Kleppe and Rosa Miró-Roig

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The goal of this paper is to study irreducible families Wt,tt1(b¯;a¯) of codimension 4, arithmetically Gorenstein schemes Xn defined by the submaximal minors of a t×t homogeneous matrix A whose entries are homogeneous forms of degree ajbi. 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 YY2(s) where s= deg(detA) and Yn is a codimension-2, arithmetically Cohen–Macaulay scheme defined by the maximal minors of the matrix obtained deleting a suitable row of A.

Article information

Algebra Number Theory, Volume 3, Number 4 (2009), 367-392.

Received: 3 October 2007
Revised: 12 December 2008
Accepted: 12 December 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M12: Determinantal varieties [See also 13C40] 14C05: Parametrization (Chow and Hilbert schemes) 14H10: Families, moduli (algebraic) 14J10: Families, moduli, classification: algebraic theory
Secondary: 14N05: Projective techniques [See also 51N35]

Hilbert scheme arithmetically Gorenstein determinantal schemes


Kleppe, Jan; Miró-Roig, Rosa. Ideals generated by submaximal minors. Algebra Number Theory 3 (2009), no. 4, 367--392. doi:10.2140/ant.2009.3.367.

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