Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A cone theorem for nef curves

Author: Brian Lehmann
Journal: J. Algebraic Geom. 21 (2012), 473-493
Published electronically: November 2, 2011
MathSciNet review: 2914801
Full-text PDF

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 [Enhancements On Off] (What's this?)

  • Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI
  • C. Araujo, The cone of effective divisors of log varieties after Batyrev, 2005, arXiv:math/0502174v1.
  • Carolina Araujo, The cone of pseudo-effective divisors of log varieties after Batyrev, Math. Z. 264 (2010), no. 1, 179–193. MR 2564937, DOI
  • S. Barkowski, The cone of moving curves of a smooth Fano-threefold, 2007, arXiv:math/0703025.
  • Victor V. Batyrev, The cone of effective divisors of threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 337–352. MR 1175891
  • 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.
  • Th. Bauer, A. Küronya, and T. Szemberg, Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math. 576 (2004), 209–233. MR 2099205, DOI
  • Alexandr Borisov, Boundedness theorem for Fano log-threefolds, J. Algebraic Geom. 5 (1996), no. 1, 119–133. MR 1358037
  • Sébastien Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76 (English, with English and French summaries). MR 2050205, DOI
  • Steven D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986), no. 1, 149–156. MR 835801, DOI
  • Yujiro Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), no. 3, 603–633. MR 744865, DOI
  • János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
  • János Kollár, The cone theorem. Note to a paper: “The cone of curves of algebraic varieties” [Ann. of Math. (2) 119 (1984), no. 3, 603–633; MR0744865 (86c:14013b)] by Y. Kawamata, Ann. of Math. (2) 120 (1984), no. 1, 1–5. MR 750714, DOI
  • Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
  • Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
  • 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

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.