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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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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Additional Information

M. Brodmann
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland
Email: markus.brodmann@math.uzh.ch

E. Park
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea
Email: euisungpark@korea.ac.kr

P. Schenzel
Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany
Email: peter.schenzel@informatik.uni-halle.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10667-6
PII: S 0002-9939(2010)10667-6
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