Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T23:10:21.987Z Has data issue: false hasContentIssue false

On classification of ℚ-fano 3-folds of Gorenstein index 2. I

Published online by Cambridge University Press:  22 January 2016

Hiromichi Takagi*
Affiliation:
RIMS, Kyoto University, Kitashirakawa Sakyo-ku Kyoto, 606-8502, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We formulate a generalization of K. Takeuchi’s method to classify smooth Fano 3-folds and use it to give a list of numerical possibilities of ℚ-Fano 3-folds X with Pic X = ℤ(−2KX) and h0(−KX) ≥ 4 containing index 2 points P such that (X, P) ≃ ({xy + z2 + ua = 0}/ℤ2(1, 1, 1, 0), o) for some a ∈ ℕ. In particular we prove that then (–KX)3 ≤ 15 and h0(–KX) ≤ 10. Moreover we show that such an X is birational to a simpler Mori fiber space.

Keywords

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[Alt] Altınok, S., Graded rings corresponding to polarized K3 surfaces and ℚ-fano 3-folds, Univ. of Warwick, ph.D. thesis.Google Scholar
[Amb99] Ambro, F., Ladders on Fano varieties, J. Math. Sci., 94 (1999), 11261135.Google Scholar
[Art69] Artin, M., Algebraic approximation of structure over complete local rings, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 2358.Google Scholar
[AW93] Andreatta, M. and Wiśniewski, J., A note on nonvanishing and applications, Duke Math. J., 72 (1993), 739755.Google Scholar
[Băd84] Bădescu, L., Hyperplane sections and deformations, Lecture Notes in Math., vol. 1056, Springer-Verlag, Berlin-New York (1984), pp. 133.Google Scholar
[CF93] Campana, F. and Flenner, H., Projective threefolds containing a smooth rational surface with ample normal bundle, J. reine angew. Math., 440 (1993), 7798.Google Scholar
[CT86] Colliot-Thélène, J.-L., Arithmétique des variétés rationnelles et problèmes bi-rationnels, Proc. Int. Conf. Math. (1986), 641653.Google Scholar
[Cut88] Cutkosky, S., Elementary contractions of Gorenstein threefolds, Math. Ann., 280 (1988), 521525.CrossRefGoogle Scholar
[Fle00] Fletcher, A. R., Working with weighted complete intersections, Explicit birational geometry of 3-folds (2000), pp. 101174.Google Scholar
[Fra91] Francia, P., On the base points of the bicanonical system, Symposia Math., 32 (1991), 141150.Google Scholar
[Fuj80] Fujita, T., On the structure of polarized manifolds with total deficiency one, part I, J. Math. Soc. of Japan, 32 (1980), 709725.Google Scholar
[Fuj81] Fujita, T., On the structure of polarized manifolds with total deficiency one, part II, J. Math. Soc. of Japan, 33 (1981), 415434.CrossRefGoogle Scholar
[Fuj84] Fujita, T., On the structure of polarized manifolds with total deficiency one, part III, J. Math. Soc. of Japan, 36 (1984), 7589.Google Scholar
[Fuj90] Fujita, T., On singular Del Pezzo varieties, Lecture Notes in Math. vol. 1417, Springer-Verlag, Berlin-New York (1990), pp. 117128.Google Scholar
[Isk77] Iskovskih, V. A., Fano 3-folds 1, Izv. Akad. Nauk SSSR Ser. Mat, 41 (1977); English transl. in Math. USSR Izv. 11 (1977), 485527.Google Scholar
[Isk78] Iskovskih, V. A., Fano 3-folds 2, Izv. Akad. Nauk SSSR Ser. Mat, 42 (1978), 506549; English transl. in Math. USSR Izv. 12 (1978), 469506.Google Scholar
[Isk79] Iskovskih, V. A., Anticanonical models of three-dimensional algebraic varieties, Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, 12 (1979), 59157; English transl. in J. Soviet. Math. 13 (1980), 745814.Google Scholar
[Isk90] Iskovskih, V. A., Double projection from a line on Fano threefolds of the first kind; English transl. in Math. USSR Sbornik 66 (1990), 265284.CrossRefGoogle Scholar
[K+92] Kollár, J. et al., Flips and abundance for algebraic threefolds, vol. 211, Astérisque, 1992.Google Scholar
[Kaw82] Kawamata, Y., A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261 (1982), 4346.Google Scholar
[Kaw86] Kawamata, Y., On the plurigenera of minimal algebraic 3-folds with K ≡ 0, Math. Ann., 275 (1986), 539546.CrossRefGoogle Scholar
[Kaw92] Kawamata, Y., Boundedness of ℚ-Fano threefolds, Proc. Int. Conf. Algebra, Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, (1992), pp. 439445.Google Scholar
[Kaw93] Kawamata, Y., The minimal discrepancy of a 3-fold terminal singularity, Russian Acad. Sci. Izv. Math., 40 (1993), 193195, Appendix to ‘3-fold log flips’.Google Scholar
[KaM98] Kawachi, T. and Maşek, V., Reider-type theorems on normal surfaces, J. Alg. Geom., 7 (1998), 239249.Google Scholar
[Kawa00] Kawachi, T., On the base point freeness of adjoint bundles on normal surfaces, Manuscripta Math., 101 (2000), 2338.CrossRefGoogle Scholar
[KM92] Kollár, J. and Mori, S., Classification of three dimensional flips, J. of Amer. Math. Soc., 5 (1992), 533703.Google Scholar
[KMM87] Kawamata, Y., Matsuda, K., and Matsuki, K., Introduction to the minimal model problem, Adv. St. Pure Math., vol. 10 (1987), pp. 287360.Google Scholar
[Kod53] Kodaira, K., On a differential method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA., 39 (1953), 12681273.Google Scholar
[Kol89] Kollár, J., Flops, Nagoya Math. J., 113 (1989), 1536.CrossRefGoogle Scholar
[Lau77] Laufer, H., On minimally elliptic singularities, Amer. Jour. Math., 99 (1977), 12571295.Google Scholar
[Luo98] Luo, T., Divisorial extremal contractions of threefolds: divisor to point, Amer. J. of Math., 120 (1998), 441451.CrossRefGoogle Scholar
[Min99] Minagawa, T., Deformations of ℚ-Calabi-Yau 3-folds and ℚ-Fano 3-folds of Fano index 1, J. Math. Sci. Univ. Tokyo, 6 (1999), no. 2, 397414.Google Scholar
[MM81] Mori, S. and Mukai, S., Classification of Fano 3-folds with b2 ≥ 2, Manuscripta Math., 36 (1981), 147162.Google Scholar
[MM83] Mori, S. and Mukai, S., On Fano 3-folds with b2 ≥ 2, Algebraic and Analytic Varieties, Adv. Stud. in Pure Math., vol. 1 (1983), pp. 101129.Google Scholar
[MM85] Mori, S. and Mukai, S., Classification of Fano 3-folds with b2 ≥ 2, I, Algebraic and Topological Theories, 1985, to the memory of Dr. Takehiko MIYATA, 496545.Google Scholar
[Mor82] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math., 116 (1982), 133176.Google Scholar
[Mor85] Mori, S., On 3-dimensional terminal singularities, Nagoya Math. J., 98 (1985), 4366.CrossRefGoogle Scholar
[Mor88] Mori, S., Flip theorem and the existence of minimal models for 3-folds, J. of Amer. Math. Soc., 1 (1988), 117253.Google Scholar
[Muk95] Mukai, S., New development of the theory of Fano threefolds: Vector bundle method and moduli problem, in Japanese, Sugaku, 47 (1995), 125144.Google Scholar
[Pro97] Prokhorov, Y., On the existence of complements of the canonical divisor for Mori conic bundles; English transl. in Sbornik: Mathematics, 188 (1997), no. 11, 16651685.Google Scholar
[Ram72] Ramanujam, C. P., Remarks on the Kodaira vanishing theorem, J. of the Indian Math. Soc., 36 (1972), 4151.Google Scholar
[Reid80] Reid, M., Lines on Fano 3-folds according to Shokurov, Tech. report, Mittag-Leffler Institute 11 (1980).Google Scholar
[Reid87] Reid, M., Young person’s guide to canonical singularities, Algebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math., vol. 46 (1987), pp. 345414.Google Scholar
[Reid94] Reid, M., Nonnormal del Pezzo surface, Publ. RIMS Kyoto Univ., 30 (1994), 695728.Google Scholar
[Reid96] Reid, M., Graded rings over K3s, abstracts of Matsumura memorial conference (1996).Google Scholar
[Sak84] Sakai, F., Weil divisors on normal surfaces, Duke Math. J., 51 (1984), no. 4, 877887.CrossRefGoogle Scholar
[San95] Sano, T., On classification of non-Gorenstein ℚ-Fano 3-folds of Fano index 1, J. Math. Soc. Japan, 47 (1995), no. 2, 369380.Google Scholar
[San96] Sano, T., Classification of non-Gorenstein ℚ-Fano d-folds of Fano index greater than d – 2, Nagoya Math. J., 142 (1996), 133143.CrossRefGoogle Scholar
[Sho79a] Shokurov, V. V., The existence of a straight line on Fano 3-folds, Izv. Akad. Nauk SSSR Ser. Mat, 43 (1979), 921963; English transl. in Math. USSR Izv. 15 (1980), 173209.Google Scholar
[Sho79b] Shokurov, V. V., Smoothness of the anticanonical divisor on a Fano 3-folds, Math. USSR. Izvestija, 43 (1979), 430441; English transl. in Math. USSR Izv. 14 (1980), 395405.Google Scholar
[Taka02] Takagi, H., On classification of ℚ-Fano 3-folds of Gorenstein index 2. II, Nagoya Math. Journal, 167 (2002), 157216.Google Scholar
[Take89] Takeuchi, K., Some birational maps of Fano 3-folds, Compositio Math., 71 (1989), 265283.Google Scholar
[Vie82] Viehweg, E., Vanishing theorems, Journ. reine angew. Math., 335 (1982), 18.Google Scholar
[Wil93] Wilson, P. M. H., The Kähler cone on Calabi-Yau threefolds (and Erratum), Invent. Math., 107, 114 (1992, 1993), 561583, 231233.Google Scholar
[Wil97] Wilson, P. M. H., Symplectic deformations of Calabi-Yau threefolds, J. Diff. Geom., 45 (1997), 611637.Google Scholar