#### 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
##### Abstract

The goal of this paper is to study irreducible families ${W}_{t,t}^{t-1}\left(\underset{¯}{b};\underset{¯}{a}\right)$ of codimension 4, arithmetically Gorenstein schemes $X\subset {ℙ}^{n}$ defined by the submaximal minors of a $t×t$ homogeneous matrix $\mathsc{A}$ whose entries are homogeneous forms of degree ${a}_{j}-{b}_{i}$. Under some numerical assumption on ${a}_{j}$ and ${b}_{i}$, we prove that the closure of ${W}_{t,t}^{t-1}\left(\underset{¯}{b};\underset{¯}{a}\right)$ is an irreducible component of ${Hilb}^{p\left(x\right)}\left({ℙ}^{n}\right)$, show that ${Hilb}^{p\left(x\right)}\left({ℙ}^{n}\right)$ is generically smooth along ${W}_{t,t}^{t-1}\left(\underset{¯}{b};\underset{¯}{a}\right)$, and compute the dimension of ${W}_{t,t}^{t-1}\left(\underset{¯}{b};\underset{¯}{a}\right)$ in terms of ${a}_{j}$ and ${b}_{i}$. To achieve these results we first prove that $X$ is determined by a regular section of ${\mathsc{ℐ}}_{Y}∕{\mathsc{ℐ}}_{Y}^{2}\left(s\right)$ where $s=deg\left(det\mathsc{A}\right)$ and $Y\subset {ℙ}^{n}$ is a codimension-2, arithmetically Cohen–Macaulay scheme defined by the maximal minors of the matrix obtained deleting a suitable row of $\mathsc{A}$.

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