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On Intrinsic Quadrics

Part of: Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  09 January 2019

Anne Fahrner  and
Jürgen Hausen
Anne Fahrner
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany Email: fahrner@math.uni-tuebingen.dejuergen.hausen@uni-tuebingen.de
Jürgen Hausen
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany Email: fahrner@math.uni-tuebingen.dejuergen.hausen@uni-tuebingen.de
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Abstract

An intrinsic quadric is a normal projective variety with a Cox ring defined by a single quadratic relation. We provide explicit descriptions of these varieties in the smooth case for small Picard numbers. As applications, we figure out in this setting the Fano examples and (affirmatively) test Fujita’s freeness conjecture.

Keywords

Fano varietyCox ringMori dream spaceintrinsic quadric

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14J10: Families, moduli, classification 14J45: Fano varieties
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 1 , February 2020 , pp. 145 - 181
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Supported by the Carl-Zeiss-Stiftung.

References

Altmann, K. and Ilten, N., Fujita’s freeness conjecture for T-varieties of complexity one, available at arxiv:1712.09927.Google Scholar
Andreatta, M., Chierici, E., and Occhetta, G., Generalized Mukai conjecture for special Fano varieties . Cent. Eur. J. Math. 2(2004), 272–293.Google Scholar
Arzhantsev, I., Derenthal, U., Hausen, J., and Laface, A., Cox rings , Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge, 2015.Google Scholar
Batyrev, V. V., On the classification of smooth projective toric varieties . Tohoku Math. J. (2) 43(1991), 569–585.Google Scholar
Berchtold, F. and Hausen, J., Cox rings and combinatorics . Trans. Amer. Math. Soc. 359(2007), 1205–1252.Google Scholar
Bonavero, L., Casagrande, C., Debarre, O., and Druel, S., Sur une conjecture de Mukai . Comment. Math. Helv. 78(2003), 601–626.Google Scholar
Bourqui, D., La conjecture de Manin géométrique pour une famille de quadriques intrinsèques . Manuscripta Math. 135(2011), 1–41 (French, with English and French summaries).Google Scholar
Casagrande, C., The number of vertices of a Fano polytope . Ann. Inst. Fourier (Grenoble) 56(2006), 121–130.Google Scholar
Cox, D. A., The homogeneous coordinate ring of a toric variety . J. Algebraic Geom. 4(1995), 17–50.Google Scholar
Casagrande, C., On the birational geometry of Fano 4-folds . Math. Ann. 355(2013), 585–628.Google Scholar
Ein, L. and Lazarsfeld, R., Global generation of pluricanonical and adjoint linear series on smooth projective threefolds . J. Amer. Math. Soc. 6(1993), 875–903.Google Scholar
Fahrner, A., Smooth Mori dream spaces of small Picard number. Doctoral Dissertation, Universität Tübingen (2017), available at https://publikationen.uni-tuebingen.de.Google Scholar
Fahrner, A., Hausen, J., and Nicolussi, M., Smooth projective varieties with a torus action of complexity 1 and Picard number 2. to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., available at arxiv:1602.04360.Google Scholar
Fujino, O., Notes on toric varieties from Mori theoretic viewpoint . Tohoku Math. J. (2) 55(2003), 551–564.Google Scholar
Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive . In: Algebraic geometry, Sendai, 1985 , 1987, pp. 167–178.Google Scholar
Hausen, J. and Herppich, E., Factorially graded rings of complexity one , Torsors, étale homotopy and applications to rational points, London Math. Soc. Lecture Note Ser., vol. 405, Cambridge Univ. Press, Cambridge, 2013, pp. 414–428.Google Scholar
Hu, Y. and Keel, S., Mori dream spaces and GIT . Michigan Math. J. 48(2000), 331–348.Google Scholar
Kawamata, Y., On Fujita’s freeness conjecture for 3-folds and 4-folds . Math. Ann. 308(1997), 491–505.Google Scholar
Kleinschmidt, P., A classification of toric varieties with few generators . Aequationes Math. 35(1988), 254–266.Google Scholar
Mori, S. and Mukai, S., Classification of Fano 3-Folds with B 2 ⩾ 2 . Manuscripta mathematica 36(1981), 147–162.Google Scholar
Mukai, S., Problems on characterization of the complex projective space . In: Birational Geometry of Algebraic Varieties, Open Problems, Katata, August 22–27 , 1988, pp. 57–60.Google Scholar
Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces . Ann. of Math. (2) 127(1988), 309–316.Google Scholar
Wiśniewski, J. A., On a conjecture of Mukai . Manuscripta Math. 68(1990), 135–141.Google Scholar
Ye, F. and Zhu, Z., On Fujita’s freeness conjecture in dimension 5(2015), available at arxiv:1511.09154.Google Scholar