On varieties of almost minimal degree II: A rank-depth formula
HTML articles powered by AMS MathViewer
- by M. Brodmann, E. Park and P. Schenzel PDF
- Proc. Amer. Math. Soc. 139 (2011), 2025-2032 Request permission
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 $\tilde X.$References
- Brodmann, M., Park, E.: On varieties of almost minimal degree I: Secant loci of rational normal scrolls. J. Pure and Applied Algebra 214 (2010), 2033 - 2043.
- Markus Brodmann and Peter Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), no. 2, 347–400. MR 2274517, DOI 10.1090/S1056-3911-06-00442-5
- Michael L. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), no. 2, 355–361. MR 1385282
- Takao Fujita, Projective varieties of $\Delta$-genus one, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 149–175. MR 1102257
- Takao Fujita, On singular del Pezzo varieties, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 117–128. MR 1040555, DOI 10.1007/BFb0083337
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
Additional Information
- M. Brodmann
- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057 Zürich, Switzerland
- MR Author ID: 41830
- 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
- MR Author ID: 155825
- ORCID: 0000-0003-1569-5100
- Email: peter.schenzel@informatik.uni-halle.de
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2025-2032
- MSC (2010): Primary 14M12; Secondary 14M05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10667-6
- MathSciNet review: 2775380