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Nonrational Weighted Hypersurfaces

Published online by Cambridge University Press:  11 January 2016

Takuzo Okada
Takuzo Okada*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, [email protected]
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Abstract

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The aim of this paper is to construct (i) infinitely many families of nonrational ℚ-Fano varieties of arbitrary dimension ≥ 4 with at most quotient singularities, and (ii) twelve families of nonrational ℚ-Fano threefolds with at most terminal singularities among which two are new and the remaining ten give an alternate proof of nonrationality to known examples. These are constructed as weighted hypersurfaces with the reduction mod p method introduced by Kollár [10].

Keywords

14E0814J4514J3014J70
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 194 , 2009 , pp. 1 - 32
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Artin, M. and Mumford, D., Some elementary examples of uniruled varieties which are not rational, Proc. London. Math. Soc, 25 (1972), 75–95.Google Scholar
[2] Brown, G. and Suzuki, K., Computing certain Fano 3-folds, Japan J. Indust. Appl. Math., 24 (2007), 241–250.CrossRefGoogle Scholar
[3] Brown, G. and Suzuki, K., Fano 3-folds with divisible anticanonical class, Manuscripta Math., 123 (2007), 37–51.Google Scholar
[4] Clemens, H. and Griffiths, P., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95 (1972), 281–356.CrossRefGoogle Scholar
[5] Corti, A., Pukhlikov, A. and Reid, M., Fano 3-fold hypersurfaces, Explicit birational geometry of 3-folds, Cambridge Univ. Press, 2000, pp. 175–258.CrossRefGoogle Scholar
[6] Dolgachev, I., Weighted projective spaces, Group actions and vector fields, Proc. Vancouver (1981), LNM 956, Springer Verlag, pp. 34–71.Google Scholar
[7] Iano-Fletcher, A. R., Working with weighted complete intersections, Explicit birational geometry of 3-folds, Cambridge Univ. Press, 2000, pp. 101–173.CrossRefGoogle Scholar
[8] Iskovskikh, V. A. and Manin, Ju. I., Three-dimensional quartics and counterexamples to the Lüroth problem, Math. USSR-Sb., 15 (1971), 815–868.Google Scholar
[9] Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, LNM 339, Springer Verlag, 1973.Google Scholar
[10] Kollar, J., Nonrational hypersurfaces, J. AMS, 8 (1995), 241–249.Google Scholar
[11] Kollar, J., Rational curves on algebraic varieties, Ergebnisse der Math. vol. 32, Springer Verlag, 1996.Google Scholar
[12] Kollar, J., Smith, K. E. and Corti, A., Rational and nearly rational varieties, Cambridge studies in advanced mathematics 92, Cambridge Univ. Press, 2004.Google Scholar
[13] Matsusaka, T., Algebraic deformations of polarized varieties, Nagoya Math. J., 31 (1968), 185–245.Google Scholar
[14] Pukhlikov, A. V., Birational automorphisms of Fano hypersurfaces, Invent. Math., 134 (1998), no. 2, 401–426.CrossRefGoogle Scholar
[15] Zhang, Q., Rational connectedness of log ℚ-Fano varieties, J. Reine Angew. Math., 590 (2006), 131–142.Google Scholar