A cone theorem for nef curves
Author:
Brian Lehmann
Journal:
J. Algebraic Geom. 21 (2012), 473-493
DOI:
https://doi.org/10.1090/S1056-3911-2011-00580-8
Published electronically:
November 2, 2011
MathSciNet review:
2914801
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Abstract |
References |
Additional Information
Abstract: Following ideas of V. Batyrev, we prove an analogue of the Cone Theorem for the closed cone of nef curves: an enlargement of the cone of nef curves is the closure of the sum of a $K_{X}$-non-negative portion and countably many $K_{X}$-negative coextremal rays. An example shows that this enlargement is necessary. We also describe the relationship between $K_{X}$-negative faces of this cone and the possible outcomes of the minimal model program.
References
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References
- V. Alexeev, Boundedness and $K^{2}$ for log surfaces, Internat. J. Math 5 (1994), no. 6, 779–810. MR 1298994 (95k:14048)
- C. Araujo, The cone of effective divisors of log varieties after Batyrev, 2005, arXiv:math/0502174v1.
- ---, The cone of pseudo-effective divisors of log varieties after Batyrev, Math. Zeit. 264 (2010), no. 1, 179–193. MR 2564937 (2010k:14018)
- S. Barkowski, The cone of moving curves of a smooth Fano-threefold, 2007, arXiv:math/0703025.
- V. Batyrev, The cone of effective divisors of threefolds, Proc. Int. Conference on Algebra, Part 3 (Novosibirsk, 1989), Cont. Math, vol. 131, Amer. Math. Soc., 1992, pp. 337–352. MR 1175891 (94f:14035)
- C. Birkar, P. Cascini, C. Hacon, and J. M$^\mathrm {c}$Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. (2010).
- S. Boucksom, J.P. Demailly, M. Pǎun, and T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, 2004, arXiv:math/0405285v1, submitted to J. Alg. Geometry.
- T. Bauer, A. Küronya, and T. Szemberg, Zariski chambers, volumes, and stable base loci, J. Reine Agnew. Math. 576 (2004), 209–233. MR 2099205 (2005h:14012)
- A. Borisov, Boundedness theorem for Fano log-threefolds, J. Alg. Geometry 5 (1996), no. 1, 119–133. MR 1358037 (96m:14058)
- S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. 37 (2004), no. 4, 45–76. MR 2050205 (2005i:32018)
- S. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986), no. 1, 149–156. MR 835801 (87f:14004)
- Y. Kawamata, The cone of curves of algebraic varieties, Ann. of Math. 119 (1984), no. 3, 603–633. MR 744865 (86c:14013b)
- J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge tracts in mathematics, vol. 134, Cambridge University Press, Cambridge UK, 1998. MR 1658959 (2000b:14018)
- J. Kollár, The cone theorem. Note to a paper: “The cone of curves of algebraic varieties”, Ann. of Math. 120 (1984), no. 1, 1–5. MR 750714 (86c:14013c)
- R. Lazarsfeld, Positivity in algebraic geometry I-II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48–49, Springer-Verlag, Berlin Heidelberg, 2004. MR 2095471 (2005k:14001a)
- N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208 (2005h:14015)
- Y.T. Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, 2006, arXiv:math/0610740v1.
- Q. Xie, The nef curve cone revisited, 2005, arXiv:math/0501193.
Additional Information
Brian Lehmann
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005
MR Author ID:
977848
Email:
blehmann@rice.edu
Received by editor(s):
July 29, 2009
Received by editor(s) in revised form:
November 23, 2010
Published electronically:
November 2, 2011
Additional Notes:
This material is based upon work supported under a National Science Foundation Graduate Research Fellowship.
