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Fano manifolds of index n − 2 and the cone conjecture

Published online by Cambridge University Press:  09 October 2017

IZZET COSKUN  and
ARTIE PRENDERGAST-SMITH
IZZET COSKUN
Affiliation:
Department of Mathematics, Statistics and CS, University of Illinois at Chicago, Chicago, IL 60607, U.S.A. e-mail: [email protected]
ARTIE PRENDERGAST-SMITH
Affiliation:
Department of Mathematical Sciences, Loughborough University, LE11 3TU. e-mail: [email protected]
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Abstract

The Morrison–Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi–Yau pair have rational, polyhedral fundamental domains. In [CPS], we proved the conjecture for certain blowups of Fano manifolds of index n - 1. In this paper, we consider the Morrison–Kawamata conjecture for blowups of Fano manifolds of index n - 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 1 - 31
Copyright
Copyright © Cambridge Philosophical Society 2017 

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Footnotes

†

Partially supported by the NSF CAREER grant DMS-0950951535.

‡

Partially supported by EPSRC First Grant EP/L026570/1.

References

REFERENCES

[AC] Araujo, C. and Castravet, A.-M.. Classification of 2-Fano manifolds with high index, in A Celebration of Algebraic Geometry (Clay Mathematics Institute, Cambridge MA., 2013).Google Scholar
[BCHM] Birkar, C., Cascini, P., Hacon, C. and McKernan, J.. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468.Google Scholar
[Br] Brav, C. Sage code to compute fundamental domains. Available from http://homepages.math.uic.edu/~coskun/Fanosage.txt.Google Scholar
[CO] Cantat, S. and Oguiso, K. Birational automorphism group and the movable cone theorem for Calabi–Yau manifolds of Wehler type via universal Coxeter groups. Amer. J. Math. 137 (2015), no. 4, 1013–1044.Google Scholar
[CPS] Coskun, I. and Prendergast–Smith, A. Fano manifolds of index n-1 and the cone conjecture. Int. Math. Res. Not. no. 9 (2014), 2401–2439.Google Scholar
[F] Fujita, T. Classification of projective varieties of Δ-genus one. Proc. Japan Acad. Math. Sci. Ser. A 58, no. 3 (1982), 113–116.Google Scholar
[GLP] Galluzzi, F., Lombardo, G. and Peters, C. Automorphs of indefinite quadratic binary forms and K3 surfaces with Picard number 2. Rend. Semin. Ma. Univ. Politec. Torino 68 (2010), no. 1, 57–77.Google Scholar
[GP] Gruson, L. and Peskine, Ch.. Courbes de l'espace projectif: variéteés de sécantes. In Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981), pp. 1–31. Progr. Math. 24 (Birkäuser, Boston, Mass., 1982).Google Scholar
[Ha] Hartshorne, R. Algebraic Geometry (Springer 1977).Google Scholar
[IM] Iskovskikh, V.A. and Manin, Yu. I.. Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sb. 86 (1971), 140–166.Google Scholar
[IP] Iskovskikh, V.A. and Prokhorov, Yu. G.. Algebraic Geometry V, Fano Varieties. Encyclopedia Math. Sci. Vol. 47 (Springer, Berlin 1999).Google Scholar
[Ka1] Kawamata, Y. On the length of an extremal rational curve. Invent. Math. 105 (1991), no. 3, 609–611.Google Scholar
[Ka2] Kawamata, Y. On the cone of divisors of Calabi–Yau fiber spaces. Internat. J. Math. 8 (1997), no. 5, 665–687.Google Scholar
[KO] Kobayashi, S. and Ochiai, T. Characterisations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47.Google Scholar
[Ko] Kollár, J. Rational Curves on Algebraic Varieties (Springer, 1996).Google Scholar
[KM] Kollár, J. and Mori, S.. Birational Geometry of Algebraic Varieties (Cambridge University Press, 1998).Google Scholar
[LPS] Lazić, V. and Prendergast–Smith, A. Effective Calabi–Yau pairs of Picard rank 2. Preprint, available from Available from http://homepages.math.uic.edu/~coskun/LPS.pdf.Google Scholar
[Lo] Looijenga, E. Discrete automorphism groups of convex cones of finite type. Compositio. Math. 150, no. 11 (2014), 1939–1962.Google Scholar
[Mo] Morrison, D. Compactifications of moduli spaces inspired by mirror symmetry. Journées de géométrie algébrique d'Orsay (Orsay, 1992). Astérisque 218 (1993), 243–271.Google Scholar
[Mu] Mukai, S. Birational classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86, no. 9 (1992), 3000–3002.Google Scholar
[Na] Namikawa, Y. Periods of Enriques surfaces. Math. Ann. 270 (1985), 201–222.Google Scholar
[O] Oguiso, K. On the finiteness of fiber-space structures on a Calabi–Yau 3-fold. J. Math. Sci. 106 (2001), 3320–3335.Google Scholar
[PS1] Prendergast–Smith, A. The cone conjecture for some rational elliptic threefolds. Math. Z. 272 (2012), no. 1–2, 589–605.Google Scholar
[PS2] Prendergast-Smith, A. The cone conjecture for abelian varieties. J. Math. Sci. Univ. Tokyo 19 (2012), no. 2, 243–261.Google Scholar
[RS] Ravindra, G. V. and Srinivas, V. The Noether–Lefschetz theorem for the divisor class group. J. Algebra 322 (2009), no. 9, 3373–3391.Google Scholar
[Sa] Stein, W. A. et al. SageMathCloud Software, The Sage Development Team (2015), http://cloud.sagemath.com.Google Scholar
[St] Sterk, H. Finiteness results for algebraic K3 surfaces. Math. Z. 189 (1985), no. 4, 507–513.Google Scholar
[Sz] Szendröi, B. Some finiteness results for Calabi–Yau threefolds. J. London Math Soc. 60 (1999), 689–699.Google Scholar
[SW] Szurek, M. and Wiśniewski, J. A. Fano bundles over P3 and Q 3. Pacific J. Math. 141, no. 1 (1990), 197–208.Google Scholar
[To1] Totaro, B. Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves. Comp. Math. 144 (2008), no. 5, 1176–1198.Google Scholar
[To2] Totaro, B. The cone conjecture for Calabi–Yau pairs in dimension 2. Duke Math. J. 154 (2010), no. 2, 241–263.Google Scholar
[Ue] Uehara, H. Calabi–Yau threefolds with infinitely many divisorial contractions. J. Math. Kyoto Univ. 44 (2004), 99–118.Google Scholar
[Vi] Vial, C. Algebraic cycles and fibrations. Doc. Math. 18 (2013), 1521–1553.Google Scholar
[W] Wilson, P.M.H. Minimal models of Calabi–Yau threefolds. Classification of Algebraic Varieties (L'Aquila, 1992), 403–410.Google Scholar
[Wi] Wiśniewski, J. On Fano manifolds of large index. Manuscripta Math. 70 (1991), 145–152.Google Scholar