Non-elementary Fano conic bundles

• Autores: Emmanuel A. Romano
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 1, 2019, págs. 33-50
• Idioma: inglés
• DOI: 10.1007/s13348-018-0218-x
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundles f:X\rightarrow Y. Denoting by \rho _{X} the Picard number of X, we investigate such contractions when \rho _{X}-\rho _{Y}>1, called non-elementary. We prove that \rho _{X}-\rho _{Y}\le 8, and we deduce new geometric information about our varieties X and Y, depending on \rho _{X}-\rho _{Y}. Using our results, we show that some known examples of Fano conic bundles are elementary. Moreover, when we allow that X is locally factorial with canonical singularities and with at most finitely many non-terminal points, and f:X\rightarrow Y is a fiber type K_{X}-negative contraction with one-dimensional fibers, we show that \rho _{X}-\rho _{Y}\le 9.

• Referencias bibliográficas
  • Alexeev, V.: Higher dimensional analogues of stable curves. In: Proceedings of Madrid ICM2006, European Mathematical Society Publishing House,...
  • Ando, T.: On extremal rays of the higher-dimensional varieties. Invent. Math. 81, 347–357 (1985)
  • Beauville, A.: Varieties de Prym et Jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. 10, 309–391 (1977)
  • Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405–468...
  • Casagrande, C.: Quasi-elementary contractions of Fano manifolds. Compos. Math. 144, 1429–1460 (2008)
  • Casagrande, C.: On Fano manifolds with a birational contraction sending a divisor to a curve. Mich. Math. J. 58, 783–805 (2009)
  • Casagrande, C.: On the Picard number of divisors in Fano manifolds. Ann. Sci. École Norm. Sup. 45, 363–403 (2012)
  • Casagrande, C.: On some Fano manifolds admitting a rational fibration. J. Lond. Math. Soc. 90, 1–28 (2014)
  • Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)
  • Della Noce, G.: On the Picard number of singular Fano varieties. Int. Math. Res. Not. 4, 955–990 (2014)
  • Ejiri, S.: Positivity of anti-canonical divisors and F-purity of fibers. arXiv:1604.02022v2 (2016)
  • Fujino, O., Gongyo, Y.: On images of weak Fano manifolds. Mathematische Zeitschrift 270(1–2), 531–544 (2012)
  • Hartshorne, R.: Algebraic Geometry. Springer, New York (1997)
  • Hu, Y., Keel, S.: Mori dream space and GIT. Mich. Math. J. 48, 331–348 (2000)
  • Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge...
  • Kollár, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differ. Geom. 36(3), 765–779 (1992)
  • Lazarsfeld, R.: Positivity in Algebraic Geometry I. Ergebn. Math. Grenzg. Springer, Berlin (2004)
  • Mori, S., Mukai, S.: Classification of Fano 3-folds with b_{2}\geqslant 2. Manuscr. Math. 36, 147–162 (1981). Erratum, Manuscr. Math. 110,...
  • Mori, S., Mukai, S.: On Fano 3-folds with b_{2}\geqslant 2. Adv. Stud. Pure Math. 1, 101–129 (1983)
  • Mori, S., Mukai, S.: Classifications of Fano 3-folds with b_{2}\geqslant 2, I. In: Algebraic and topological theories. Papers from the symposium...
  • Occhetta, G.: A characterization of products of projective spaces. Can. Math. Bull. 49, 270–280 (2006)
  • Parshin, A.N., Shafarevich, I.R.: Algebraic Geometry V: Fano Varieties. Encyclopedia of Mathematical Sciences. Springer, Berlin (1999)
  • Sarkisov, V.G.: On conic bundle structures. Math. USSR Izv. 20, 355–390 (1983)
  • Wiśniewski, J.A.: On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417, 141–157 (1991)