Blow-ups and Fano manifolds of large pseudoindex

  • Carla Novelli

Abstract

We describe the Kleiman-Mori cones of Fano manifolds of large pseudoindex that admit a structure of smooth blow-up.

References

Andreatta, M., Chierici, E., and Occhetta, G., Generalized Mukai conjecture for special Fano varieties, Cent. Eur. J. Math. 2 (2004), no. 2, 272–293. https://doi.org/10.2478/BF02476544

Andreatta, M. and Occhetta, G., Special rays in the Mori cone of a projective variety, Nagoya Math. J. 168 (2002), 127–137. https://doi.org/10.1017/S0027763000008400

Andreatta, M. and Occhetta, G., Fano manifolds with long extremal rays, Asian J. Math. 9 (2005), no. 4, 523–543. https://doi.org/10.4310/AJM.2005.v9.n4.a5

Beltrametti, M. C., Sommese, A. J., and Wiśniewski, J. A., Results on varieties with many lines and their applications to adjunction theory, in “Complex algebraic varieties (Bayreuth, 1990)'', Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 16--38. https://doi.org/10.1007/BFb0094508

Bonavero, L., Casagrande, C., Debarre, O., and Druel, S., Sur une conjecture de Mukai, Comment. Math. Helv. 78 (2003), no. 3, 601–626. https://doi.org/10.1007/s00014-003-0765-x

Campana, F., Coréduction algébrique d'un espace analytique faiblement Kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223. https://doi.org/10.1007/BF01393876

Chierici, E. and Occhetta, G., The cone of curves of Fano varieties of coindex four, Internat. J. Math. 17 (2006), no. 10, 1195–1221. https://doi.org/10.1142/S0129167X06003850

Chierici, E. and Occhetta, G., Fano manifolds and blow-ups of low-dimensional subvarieties, J. Korean Math. Soc. 47 (2010), no. 1, 189–213. https://doi.org/10.4134/JKMS.2010.47.1.189

Cho, K., Miyaoka, Y., and Shepherd-Barron, N. I., Characterizations of projective space and applications to complex symplectic manifolds, in “Higher dimensional birational geometry (Kyoto, 1997)'', Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1--88.

Kobayashi, S. and Ochiai, T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. https://doi.org/10.1215/kjm/1250523432

Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 32, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/978-3-662-03276-3

Kollár, J., Miyaoka, Y., and Mori, S., Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765–779.

Mori, S., Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. https://doi.org/10.2307/1971241

Mukai, S., Problems on characterization of the complex projective space, in “Birational geometry of algebraic varieties: open problems” (Mori, S., Kollár, J., Mukai, S., and Miyaoka, Y., eds.), The 23rd International Symposium, August 22-27, 1988, Katata, Division of Mathematics, The Taniguichi Foundation, 1988, pp. 57--60.

Novelli, C., On Fano manifolds with an unsplit dominating family of rational curves, Kodai Math. J. 35 (2012), no. 3, 425–438. https://doi.org/10.2996/kmj/1352985447

Novelli, C., On Fano manifolds of large pseudoindex, J. Algebra 449 (2016), 138–162. https://doi.org/10.1016/j.jalgebra.2015.10.012

Novelli, C. and Occhetta, G., Ruled Fano fivefolds of index two, Indiana Univ. Math. J. 56 (2007), no. 1, 207–241. https://doi.org/10.1512/iumj.2007.56.2905

Novelli, C. and Occhetta, G., Rational curves and bounds on the Picard number of Fano manifolds, Geom. Dedicata 147 (2010), 207–217. https://doi.org/10.1007/s10711-009-9452-4

Novelli, C. and Occhetta, G., Manifolds covered by lines and extremal rays, Canad. Math. Bull. 55 (2012), no. 4, 799–814. https://doi.org/10.4153/CMB-2011-119-7

Occhetta, G., A characterization of products of projective spaces, Canad. Math. Bull. 49 (2006), no. 2, 270–280. https://doi.org/10.4153/CMB-2006-028-3

Wiśniewski, J. A., On a conjecture of Mukai, Manuscripta Math. 68 (1990), no. 2, 135–141. https://doi.org/10.1007/BF02568756

Wiśniewski, J. A., On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141–157. https://doi.org/10.1515/crll.1991.417.141
Published
2019-01-13
How to Cite
Novelli, C. (2019). Blow-ups and Fano manifolds of large pseudoindex. MATHEMATICA SCANDINAVICA, 124(1), 34-50. https://doi.org/10.7146/math.scand.a-109996
Section
Articles