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On Intrinsic Quadrics

  • Anne Fahrner (a1) and Jürgen Hausen (a1)

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.

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Supported by the Carl-Zeiss-Stiftung.

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[1] Altmann, K. and Ilten, N., Fujita’s freeness conjecture for T-varieties of complexity one, available at arxiv:1712.09927.
[2] Andreatta, M., Chierici, E., and Occhetta, G., Generalized Mukai conjecture for special Fano varieties . Cent. Eur. J. Math. 2(2004), 272293.
[3] Arzhantsev, I., Derenthal, U., Hausen, J., and Laface, A., Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge, 2015.
[4] Batyrev, V. V., On the classification of smooth projective toric varieties . Tohoku Math. J. (2) 43(1991), 569585.
[5] Berchtold, F. and Hausen, J., Cox rings and combinatorics . Trans. Amer. Math. Soc. 359(2007), 12051252.
[6] Bonavero, L., Casagrande, C., Debarre, O., and Druel, S., Sur une conjecture de Mukai . Comment. Math. Helv. 78(2003), 601626.
[7] Bourqui, D., La conjecture de Manin géométrique pour une famille de quadriques intrinsèques . Manuscripta Math. 135(2011), 141 (French, with English and French summaries).
[8] Casagrande, C., The number of vertices of a Fano polytope . Ann. Inst. Fourier (Grenoble) 56(2006), 121130.
[9] Cox, D. A., The homogeneous coordinate ring of a toric variety . J. Algebraic Geom. 4(1995), 1750.
[10] Casagrande, C., On the birational geometry of Fano 4-folds . Math. Ann. 355(2013), 585628.
[11] Ein, L. and Lazarsfeld, R., Global generation of pluricanonical and adjoint linear series on smooth projective threefolds . J. Amer. Math. Soc. 6(1993), 875903.
[12] Fahrner, A., Smooth Mori dream spaces of small Picard number. Doctoral Dissertation, Universität Tübingen (2017), available at
[13] 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.
[14] Fujino, O., Notes on toric varieties from Mori theoretic viewpoint . Tohoku Math. J. (2) 55(2003), 551564.
[15] Fujita, T., On polarized manifolds whose adjoint bundles are not semipositive . In: Algebraic geometry, Sendai, 1985, 1987, pp. 167178.
[16] 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. 414428.
[17] Hu, Y. and Keel, S., Mori dream spaces and GIT . Michigan Math. J. 48(2000), 331348.
[18] Kawamata, Y., On Fujita’s freeness conjecture for 3-folds and 4-folds . Math. Ann. 308(1997), 491505.
[19] Kleinschmidt, P., A classification of toric varieties with few generators . Aequationes Math. 35(1988), 254266.
[20] Mori, S. and Mukai, S., Classification of Fano 3-Folds with B 2 ⩾ 2 . Manuscripta mathematica 36(1981), 147162.
[21] Mukai, S., Problems on characterization of the complex projective space . In: Birational Geometry of Algebraic Varieties, Open Problems, Katata, August 22–27, 1988, pp. 5760.
[22] Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces . Ann. of Math. (2) 127(1988), 309316.
[23] Wiśniewski, J. A., On a conjecture of Mukai . Manuscripta Math. 68(1990), 135141.
[24] Ye, F. and Zhu, Z., On Fujita’s freeness conjecture in dimension 5(2015), available at arxiv:1511.09154.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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