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Canonical Toric Fano Threefolds

  • Alexander M. Kasprzyk (a1)

Abstract

An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

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[Bat81] Batyrev, V. V., Toric Fano threefolds. Izv. Akad. Nauk SSSR Ser. Mat. 45(1981), no. 4, 704–717, 927.
[Bat91] Batyrev, V. V., On the classification of smooth projective Toric varieties. Tohoku Math. J. 43(1991), no. 4, 569–585. doi:10.2748/tmj/1178227429
[Bat94] Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3(1994), no. 3, 493–535.
[Bat99] Batyrev, V. V., On the classification of toric Fano 4-folds. J. Math. Sci. (New York) 94(1999), no. 1, 1021–1050. doi:10.1007/BF02367245
[BB] Borisov, A. A. and Borisov, L. A., Three-dimensional toric Fano varieties with terminal singularities, unpublished, text in Russian, available upon request.
[BB92] Borisov, A. A. and Borisov, L. A., Singular toric Fano three-folds. (Russian) Mat. Sb. 183(1992), no. 2, 134–141, translation in: Russian Acad. Sci. Sb. Math. 75(1993), no. 1, 277–283.
[BB96] Batyrev, V. V. and Borisov, L. A., On Calabi-Yau complete intersections in toric varieties. In: Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 39–65.
[Bor00] Borisov, A. A., Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint. http://arxiv.org/abs/math/0001109
[Bro07] Brown, G., A database of polarized K3 surfaces. Experiment. Math. 16(2007), no. 1, 7–20.
[Con02] Conrads, H., Weighted projective spaces and reflexive simplices. Manuscripta Math. 107(2002), no. 2, 215–227. doi:10.1007/s002290100235
[Cut89] Cutkosky, S. D., On Fano 3-folds. Manuscripta Math. 64(1989), no. 2, 189–204. doi:10.1007/BF01160118
[Dan78] Danilov, V. I., The geometry of toric varieties. Uspekhi Mat. Nauk 33(1978), no. 2(200), 85–134, 247.
[FS04] Fujino, O. and Sato, H., Introduction to the toric Mori theory. Michigan Math. J. 52(2004), no. 3, 649–665. doi:10.1307/mmj/1100623418
[Ful93] Fulton, W., Introduction to toric varieties. Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.
[Isk79a] Iskovskih, V. A., Anticanonical models of three-dimensional algebraic varieties. (Russian) Current problems in mathematics, 12, VINITI, Moscow, 1979, pp. 59–157, 239 (loose errata).
[Isk79b] Iskovskih, V. A., Birational automorphisms of three-dimensional algebraic varieties. (Russian) Current problems in mathematics, 12, VINITI, Moscow, 1979, pp. 159–236, 239 (loose errata).
[Kas06a] Kasprzyk, A. M., Toric Fano three-folds with terminal singularities. Tohoku Math. J. (2) 58(2006), no. 1, 101–121. doi:10.2748/tmj/1145390208
[Kas06b] Kasprzyk, A. M., Toric Fano varieties and convex polytopes, Ph.D. thesis, University of Bath (2006), available from http://hdl.handle.net/10247/458
[Kas08] Kasprzyk, A. M., Bounds on fake weighted projective space. Kodai Math. J. 32(2009), no. 2, 197–208. doi:10.2996/kmj/1245982903
[K M M92] Kollár, J., Miyaoka, Y., and Mori, S., Rational connectedness and boundedness of Fano manifolds. J. Differential Geom. 36(1992), no. 3, 765–779.
[KN07] Kreuzer, M. and Nill, B., Classification of toric Fano 5-folds. Adv. Geom. 9(2009), no. 1, 85–97. doi:10.1515/ADVGEO M.2009.005
[KS97] Kreuzer, M. and Skarke, H., On the classification of reflexive polyhedra. Comm. Math. Phys. 185(1997), no. 2, 495–508. doi:10.1007/s002200050100
[KS98] Kreuzer, M. and Skarke, H., Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2(1998), no. 4, 853–871.
[KS00] Kreuzer, M. and Skarke, H., Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4(2000), no. 6, 1209–1230.
[KS02] Kreuzer, M. and Skarke, H., Reflexive polyhedra, weights and toric Calabi-Yau fibrations. Rev. Math. Phys. 14(2002), no. 4, 343–374. doi:10.1142/S0129055X0200120X
[KS04] Kreuzer, M. and Skarke, H., PALP, a package for analyzing lattice polytopes with applications to toric geometry. Computer Phys. Comm. 157(2004), no. 1, 87–106. doi:10.1016/S0010-4655(03)00491-0
[M M04] Mori, S. and Mukai, S., Extremal rays and Fano 3-folds. In: The Fano Conference, Univ. Torino, Turin, 2004, pp. 37–50.
[MU83] Mukai, S. and Umemura, H., Minimal rational threefolds. In: Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., 1016, Springer, Berlin, 1983, pp. 490–518.
[Nil05] Nill, B., Gorenstein toric Fano varieties. Manuscripta Math. 116(2005), no. 2, 183–210. doi:10.1007/s00229-004-0532-3
[Nil06] Nill, B., Classification of pseudo-symmetric simplicial reflexive polytopes. In: Algebraic and geometric combinatorics, Contemp. Math., 423, American Mathematical Society, Providence, RI, 2006, pp. 269–282.
[Øbr07] Øbro, Mikkel, An algorithm for the classification of smooth Fano polytopes. http://arxiv.org/abs/0704.0049.
[Oda78] Oda, T., Torus embeddings and applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 57, Springer-Verlag, Berlin-New York, 1978.
[Pro05] Prokhorov, Yu. G., The degree of Fano threefolds with canonical Gorenstein singularities. (Russian) Mat. Sb. 196(2005), no. 1, 81–122.
[PRV00] Poonen, B. and F. Rodriguez-Villegas, Lattice polygons and the number 12. Amer. Math. Monthly 107(2000), no. 3, 238–250. doi:10.2307/2589316
[Rei83] Reid, M., Minimal models of canonical 3-folds. In: Algebraic varieties and analytic varieties, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, pp. 131–180.
[Sat00] Sato, Hiroshi, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. (2) 52(2000), no. 3, 383–413. doi:10.2748/tmj/1178207820
[Šok79] Šokurov, V. V., The existence of a line on Fano varieties. Izv. Akad. Nauk SSSR Ser. Mat. 43(1979), no. 4, 922–964, 968.
[Tak89] Takeuchi, K., Some birational maps of Fano 3-folds. Compositio Math. 71(1989), no. 3, 265–283.
[Wiś02] Wiśniewski, J. A., Toric Mori theory and Fano manifolds. In: Geometry of toric varieties, Sémin. Congr., 6, Soc. Math. France, Paris, 2002, pp. 249–272.
[WW82] Watanabe, K. and Watanabe, M., The classification of Fano 3-folds with torus embeddings. Tokyo J. Math. 5(1982), no. 1, 37–48. doi:10.3836/tjm/1270215033
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Canonical Toric Fano Threefolds

  • Alexander M. Kasprzyk (a1)

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