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On varieties of almost minimal degree II: A rank-depth formula

Authors: M. Brodmann, E. Park and P. Schenzel
Journal: Proc. Amer. Math. Soc. 139 (2011), 2025-2032
MSC (2010): Primary 14M12; Secondary 14M05
Published electronically: November 24, 2010
MathSciNet review: 2775380
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Abstract: Let $ X \subset \mathbb{P}^r_K$ denote a variety of almost minimal degree other than a normal del Pezzo variety. Then $ X$ is the projection of a rational normal scroll $ \tilde X \subset {\mathbb{P}}^{r+1}_K$ from a point $ p \in {\mathbb{P}}^{r+1}_K \setminus \tilde X.$ We show that the arithmetic depth of $ X$ can be expressed in terms of the rank of the matrix $ M'(p),$ where $ M'$ is the matrix of linear forms whose $ 3\times 3$ minors define the secant variety of $ \tilde X.$

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M. Brodmann
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland

E. Park
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea

P. Schenzel
Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany

Keywords: Variety of almost minimal degree, depth formula, secant cone
Received by editor(s): February 6, 2010
Received by editor(s) in revised form: June 10, 2010
Published electronically: November 24, 2010
Communicated by: Irena Peeva
Article copyright: © Copyright 2010 American Mathematical Society

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