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On the number of minimal models of a log smooth threefold. (English) Zbl 1307.14057

The minimal model of a complex projective surface is smooth and unique. For varieties of higher dimensions it is conjectured that the number of minimal models is finite up to isomorphisms. The conjecture was proved for projective varieties of general type [C. Birkar et al., J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)] and for threefolds of positive Kodaira dimension [Y. Kawamata, Int. J. Math. 8, No. 5, 665–687 (1997; Zbl 0931.14022)]. Moreover, the relative Cone Conjecture of Morrison and Kawamata [Zbl 0931.14022] and the Shokurov’s log geography, which assumes the full Minimal Model Program, imply the conjecture.
For varieties of dimension three the minimal models are no longer smooth, but P. Cascini and D.-Q. Zhang [Ann. Inst. Fourier 64, No. 1, 127–144 (2014; Zbl 1364.14012)] provided a topological bound on the index of the singularities.
In the paper under review, the authors prove that given a log smooth projective surface \((X, \sum_{i=1}^p S_i)\), where \(S_1, \dots, S_p\) are distinct prime divisors, and \(\epsilon\) a positive rational number, the number of minimal models of the pair \((X, B)\), where \(B= \sum_{i=1}^p\delta_i S_i\) and \(\delta_i \in [\epsilon, 1- \epsilon]\) for all \(i\), is bounded by a constant \(M\) depending only on the topology of \(X\) and of \(S_1, \dots, S_p\) and on \(\epsilon\). This result provides an effective version of the finite generation of adjoint rings in the two-dimensional case and generalises some previous results found in [Zbl 1364.14012]. Furthermore, the authors provide an analogous topological bound for minimal models of a non-uniruled terminal threefold pair \((X, B)\), such that the prime divisors contained in the support of \(B\) span Div\(_{\mathbb{R}}(X)\) up to numerical equivalence.
The proof uses techniques from the three-dimensional MMP: in particular Shokurov’s log geography and the proof of termination of threefold flips.

MSC:

14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
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References:

[1] Alexeev, V.; Hacon, C.; Kawamata, Y., Termination of (many) 4-dimensional log flips, Invent. Math., 168, 2, 433-448 (2007) · Zbl 1118.14017
[2] Birkar, C.; Cascini, P.; Hacon, C.; \(M^c\) Kernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23, 2, 405-468 (2010) · Zbl 1210.14019
[3] Boucksom, S.; Cacciola, S.; Lopez, A. F., Augmented base loci and restricted volumes on normal varieties · Zbl 1348.14017
[4] Cascini, P.; Lazić, V., New outlook on the Minimal Model Program, I, Duke Math. J., 161, 12, 2415-2467 (2012) · Zbl 1261.14007
[5] Corti, A.; Lazić, V., New outlook on the Minimal Model Program, II, Math. Ann., 356, 2, 617-633 (2013) · Zbl 1273.14033
[6] Corti, A., Three equivalent conjectures on the birational geometry of Fano 3-folds, Tr. Mat. Inst. Steklova, 264, 52-54 (2009)
[7] Cascini, P.; Zhang, D.-Q., Effective finite generation for adjoint rings, Ann. Inst. Fourier (2013), in press
[8] Hacon, C. D.; \(M^c\) Kernan, J., The Sarkisov program, J. Algebraic Geom., 22, 2, 389-405 (2013) · Zbl 1267.14024
[9] Kawamata, Y., On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math., 8, 5, 665-687 (1997) · Zbl 0931.14022
[10] Kaloghiros, A.-S.; Küronya, A.; Lazić, V., Finite generation and geography of models, in: Minimal Models and Extremal Rays, Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo, in press
[11] Kollár, J.; Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134 (1998), Cambridge University Press
[12] Kollár, J., Higher direct images of dualizing sheaves I, Ann. Math., 123, 11-42 (1986) · Zbl 0598.14015
[13] Kollár, J., Effective base point freeness, Math. Ann., 296, 4, 595-605 (1993) · Zbl 0818.14002
[14] Kollár, J., Shafarevich Maps and Automorphic Forms, (M.B. Porter Lectures) (1995), Princeton University Press: Princeton University Press Princeton, NJ
[15] Lazarsfeld, R., Positivity in Algebraic Geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vols. 48, 49 (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1066.14021
[16] Mori, S., On 3-dimensional terminal singularities, Nagoya Math. J., 98, 43-66 (1985) · Zbl 0589.14005
[17] Prokhorov, Y.; Shokurov, V., Towards the second main theorem on complements, J. Algebraic Geom., 18, 1, 151-199 (2009) · Zbl 1159.14020
[18] Reid, M., Young person’s guide to canonical singularities, (Algebraic Geometry. Algebraic Geometry, Bowdoin, 1985. Algebraic Geometry. Algebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math., vol. 46 (1987), Amer. Math. Soc.), 345-414
[19] Shokurov, V. V.; Choi, S. R., Geography of log models: theory and applications, Cent. Eur. J. Math., 9, 3, 489-534 (2011) · Zbl 1234.14014
[20] Shokurov, V. V., 3-fold log models, J. Math. Sci., 81, 3, 2667-2699 (1996) · Zbl 0873.14014
[21] Totaro, B., The cone conjecture for Calabi-Yau pairs in dimension 2, Duke Math. J., 154, 2, 241-263 (2010) · Zbl 1203.14046
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