Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Arithmetic properties of projective varieties of almost minimal degree


Authors: Markus Brodmann and Peter Schenzel
Journal: J. Algebraic Geom. 16 (2007), 347-400
DOI: https://doi.org/10.1090/S1056-3911-06-00442-5
Published electronically: October 11, 2006
MathSciNet review: 2274517
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Abstract: We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely $ 2$. We notably show, that such a variety $ X \subset {\mathbb{P}}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $ \tilde {X} \subset {\mathbb{P}}^{r + 1}$ from an appropriate point $ p \in {\mathbb{P}}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $ X$ by means of the projection $ \tilde {X} \rightarrow X$.

If $ X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $ B$ of the projecting variety $ \tilde {X}$ is the endomorphism ring of the canonical module $ K(A)$ of the homogeneous coordinate ring $ A$ of $ X.$ If $ X$ is non-normal and is maximally Del Pezzo, that is, arithmetically Cohen-Macaulay but not arithmetically normal, $ B$ is just the graded integral closure of $ A.$ It turns out, that the geometry of the projection $ \tilde {X} \rightarrow X$ is governed by the arithmetic depth of $ X$ in any case.

We study, in particular, the case in which the projecting variety $ \tilde {X} \subset {\mathbb{P}}^{r + 1}$ is a (cone over a) rational normal scroll. In this case $ X$ is contained in a variety of minimal degree $ Y \subset {\mathbb{P}}^r$ such that $ \operatorname{codim}_Y(X) = 1$. We use this to approximate the Betti numbers of $ X$.

In addition, we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of $ \Delta $-genus $ 1$.


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Additional Information

Markus Brodmann
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Schwitzerland
Email: brodmann@math.unizh.ch

Peter Schenzel
Affiliation: Martin-Luther-Universität Halle-Wittenberg, Institut Für Informatik, Von- Seckendorff-Platz 1, D-06120 Halle (Saale), Germany
Email: schenzel@informatik.uni-halle.de

DOI: https://doi.org/10.1090/S1056-3911-06-00442-5
Received by editor(s): August 10, 2005
Received by editor(s) in revised form: December 12, 2005
Published electronically: October 11, 2006
Additional Notes: The second author was partially supported by the Swiss National Science Foundation (Projects No. 20-66980-01 and No. 20-103491/1)

American Mathematical Society