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\(\mathbb{Q}\)-Fano threefolds of index 7. (English. Russian original) Zbl 1360.14111

Proc. Steklov Inst. Math. 294, 139-153 (2016); translation from Tr. Mat. Inst. Steklova 294, 152-166 (2016).
The author expands his previous results on \(\mathbb{Q}\)-Fano threefolds. [Y. G. Prokhorov, Sb. Math. 204, No. 3, 347–382 (2013; Zbl 1279.14051); translation from Mat. Sb. 204, No. 3, 43–78 (2013)], [Y. G. Prohorovet al., “On \(\mathbb{Q}\)-Fano 3-folds of Fano index 2”, Preprint, arXiv:1203.0852]. More precisely, he considers \(\mathbb{Q}\)-Fano threefolds \(X\) with \(\mathbb{Q}\)-Fano index \(7\). He shows that if dim \(|-K_X|\geq 15\) then \(X\) is isomorphic to one of three given varieties. [Y. G. Prokhorov, Izv. Math. 79, No. 4, 795–808 (2015; Zbl 1331.14019); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 79, No. 4, 159–174 (2015)]

MSC:

14J45 Fano varieties

Keywords:

Fano threefolds
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Full Text: DOI arXiv

References:

[1] V. Alexeev, “General elephants of Q-Fano 3-folds, ” Compos. Math. 91 (1), 91-116 (1994). · Zbl 0813.14028
[2] G. Brown, A. Kasprzyk, et al., “Graded ring database,” http://www.grdb.co.uk
[3] G. Brown, M. Kerber, and M. Reid, “Fano 3-folds in codimension 4, Tom and Jerry. Part I, ” Compos. Math. 148 (4), 1171-1194 (2012). · Zbl 1258.14049 · doi:10.1112/S0010437X11007226
[4] G. Brown and K. Suzuki, “Computing certain Fano 3-folds, ” Japan J. Ind. Appl. Math. 24 (3), 241-250 (2007). · Zbl 1166.14027 · doi:10.1007/BF03167538
[5] I. V. Dolgachev, ClassicalAlgebraic Geometry: A Modern View (Cambridge Univ. Press, Cambridge, 2012). · Zbl 1252.14001 · doi:10.1017/CBO9781139084437
[6] Flips and Abundance for Algebraic Threefolds: A Summer Seminar, Univ. Utah, Salt Lake City, 1991, Ed. by J. Kollár (Soc. Math. France, Paris, 1992), Astérisque 211. · Zbl 1166.14027
[7] M. Kawakita, “Divisorial contractions in dimension three which contract divisors to smooth points, ” Invent. Math. 145 (1), 105-119 (2001). · Zbl 1091.14007 · doi:10.1007/s002220100144
[8] M. Kawakita, “Three-fold divisorial contractions to singularities of higher indices, ” Duke Math. J. 130 (1), 57-126 (2005). · Zbl 1091.14008 · doi:10.1215/S0012-7094-05-13013-7
[9] Kawamata, Y., Boundedness of Q-Fano threefolds (1992) · Zbl 0785.14024
[10] Y. Kawamata, “The minimal discrepancy coefficients of terminal singularities in dimension 3, ” in V. V. Shokurov, “3-fold log flips,” Izv. Ross. Akad. Nauk, Ser. Mat. 56 (1), 105-203 (1992), Appendix, pp. 201-203 [Russ. Acad. Sci., Izv. Math. 40 (1), 95-202 (1993), Appendix, pp. 193-195]. · Zbl 0785.14023
[11] Kawamata, Y., Divisorial contractions to 3-dimensional terminal quotient singularities, 241-246 (1996), Berlin · Zbl 0894.14019
[12] J. Kollár and S. Mori, “Classification of three-dimensional flips, ” J. Am. Math. Soc. 5 (3), 533-703 (1992). · Zbl 0773.14004 · doi:10.2307/2152704
[13] M. Miyanishi and D. Q. Zhang, “Gorenstein log del Pezzo surfaces of rank one, ” J. Algebra 118 (1), 63-84 (1988). · Zbl 0664.14019 · doi:10.1016/0021-8693(88)90048-8
[14] S. Mori, “On a generalization of complete intersections, ” J. Math. Kyoto Univ. 15 (3), 619-646 (1975). · Zbl 0332.14019
[15] S. Mori and Yu. Prokhorov, “On Q-conic bundles, ” Publ. Res. Inst. Math. Sci. 44 (2), 315-369 (2008). · Zbl 1151.14029 · doi:10.2977/prims/1210167329
[16] Nakayama, N., The lower semi-continuity of the plurigenera of complex varieties (1987) · Zbl 0649.14003
[17] Yu. G. Prokhorov, “The degree of Q-Fano threefolds, ” Mat. Sb. 198 (11), 153-174 (2007) [Sb. Math. 198, 1683-1702 (2007)]. · Zbl 1139.14033 · doi:10.4213/sm3801
[18] Yu. Prokhorov, “Q-Fano threefolds of large Fano index. I, ” Doc. Math., J. DMV 15, 843-872 (2010). · Zbl 1218.14031
[19] Yu. G. Prokhorov, “Fano threefolds of large Fano index and large degree, ” Mat. Sb. 204 (3), 43-78 (2013) [Sb. Math. 204, 347-382 (2013)]. · Zbl 1279.14051 · doi:10.4213/sm8089
[20] Yu. G. Prokhorov, “On G-Fano threefolds, ” Izv. Ross. Akad. Nauk, Ser. Mat. 79 (4), 159-174 (2015) [Izv. Math. 79, 795-808 (2015)]. · Zbl 1331.14019 · doi:10.4213/im8349
[21] Prokhorov, Y.; Reid, M., On Q-Fano 3-folds of Fano index 2 (2016) · Zbl 1372.14033
[22] Yu. G. Prokhorov and V. V. Shokurov, “Towards the second main theorem on complements, ” J. Algebr. Geom. 18 (1), 151-199 (2009). · Zbl 1159.14020 · doi:10.1090/S1056-3911-08-00498-0
[23] V. V. Przyjalkowski and C. A. Shramov, “Double quadrics with large automorphism groups, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 167-190 (2016) [Proc. Steklov Inst. Math. 294, 154-175 (2016)]. · Zbl 1375.14146
[24] Reid, M., Young person’s guide to canonical singularities (1987) · Zbl 0634.14003
[25] V. V. Shokurov, “The nonvanishing theorem, ” Izv. Akad. Nauk SSSR, Ser. Mat. 49 (3), 635-651 (1985) [Math. USSR, Izv. 26 (3), 591-604 (1986)]. · Zbl 0623.65030
[26] K. Suzuki, “On Fano indices of Q-Fano 3-folds, ” Manuscr. Math. 114 (2), 229-246 (2004). · Zbl 1063.14049 · doi:10.1007/s00229-004-0442-4
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