Ideals generated by submaximal minors

Kleppe, Jan; Miró-Roig, Rosa

doi:10.2140/ant.2009.3.367

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Ideals generated by submaximal minors

Jan O. Kleppe and Rosa M. Miró-Roig

Vol. 3 (2009), No. 4, 367–392

DOI: 10.2140/ant.2009.3.367

Abstract

The goal of this paper is to study irreducible families $W_{t, t}^{t - 1} (\underline{b}; \underline{a})$ of codimension 4, arithmetically Gorenstein schemes $X \subset ℙ^{n}$ defined by the submaximal minors of a $t \times t$ homogeneous matrix $A$ 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 $W_{t, t}^{t - 1} (\underline{b}; \underline{a})$ is an irreducible component of ${Hilb}^{p (x)} (ℙ^{n})$ , show that ${Hilb}^{p (x)} (ℙ^{n})$ is generically smooth along $W_{t, t}^{t - 1} (\underline{b}; \underline{a})$ , and compute the dimension of $W_{t, t}^{t - 1} (\underline{b}; \underline{a})$ in terms of $a_{j}$ and $b_{i}$ . To achieve these results we first prove that $X$ is determined by a regular section of $ℐ_{Y} ∕ ℐ_{Y}^{2} (s)$ where $s = deg (det A)$ 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 $A$ .

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

Vol. 3, No. 4, 2009