A note on the fibres of Mori fibre spaces

  • Giulio Codogni
  • Andrea Fanelli
  • Roberto Svaldi
  • Luca Tasin
Research Article

Abstract

We consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high Picard rank.

Keywords

Fano varieties Mori fibre spaces Toric varieties Vertex-transitive polytopes High index High Picard rank 

Mathematics Subject Classification

14J45 14E30 14M25 14J35 14J40 

Notes

Acknowledgements

We would like to thank Cinzia Casagrande for interesting comments on this work and an anonymous referee for useful suggestions.

References

  1. 1.
    Araujo, C., Casagrande, C.: On the Fano variety of linear spaces contained in two odd-dimensional quadrics. Geom. Topol. 21(5), 3009–3045 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Batyrev, V.V.: On the classification of smooth projective toric varieties. Tohoku Math. J. (2) 43(4), 569–585 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bauer, S.: Parabolic bundles, elliptic surfaces and \({\rm SU} (2)\)-representation spaces of genus zero Fuchsian groups. Math. Ann. 290(3), 509–526 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Birkar, C.: Existence of flips and minimal models for 3-folds in char \(p\). Ann. Sci. Éc. Norm. Supér. (4) 49(1), 169–212 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23(2), 405–468 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Birkar, C., Waldron, J.: Existence of Mori fibre spaces for 3-folds in char \(p\). Adv. Math. 313, 62–101 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brown, G., Kasprzyk, A.M.: Graded ring database. http://www.grdb.co.uk/
  8. 8.
    Casagrande, C.: Contractible classes in toric varieties. Math. Z. 243(1), 99–126 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Casagrande, C.: The number of vertices of a Fano polytope. Ann. Inst. Fourier (Grenoble) 56(1), 121–130 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Casagrande, C., Codogni, G., Fanelli, A.: The blow-up of \(\mathbb{P}^4\) at 8 points and its Fano model, via vector bundles on a degree 1 del Pezzo surface (2017). arXiv:1707.09152
  11. 11.
    Castravet, A.-M.: Mori dream spaces and blow-ups (2017). arXiv:1701.04738 (to appear on the Proceedings of the AMS Summer Institute in Algebraic Geometry (University of Utah, July 2015))
  12. 12.
    Castravet, A.-M., Tevelev, J.: Hilbert’s 14th problem and Cox rings. Compos. Math. 142(6), 1479–1498 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Codogni, G., Fanelli, A., Svaldi, R., Tasin, L.: Fano varieties in Mori fibre spaces. Int. Math. Res. Not. IMRN 2016(7), 2026–2067 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  15. 15.
    Dolgachev, I.V.: On certain families of elliptic curves in projective space. Ann. Mat. Pura Appl. 183(4), 317–331 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fujita, T.: Classification of projective varieties of \({\Delta } \)-genus one. Proc. Japan Acad. Ser. A Math. Sci. 58(3), 113–116 (1982)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fujita, T.: On polarized varieties of small \({\Delta } \)-genera. Tohoku Math. J. (2) 34(3), 319–341 (1982)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gleason, I., Hubard, I.: Products of abstract polytopes (2016). arXiv:1603.03585
  19. 19.
    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry, 2nd edn. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall, Boca Raton (2004)Google Scholar
  20. 20.
    Hacon, C.D., Xu, C.: On the three dimensional minimal model program in positive characteristic. J. Amer. Math. Soc. 28(3), 711–744 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Klyachko, A.A.: Demazure models for a special class of tori. Selecta Math. Soviet. 3(1), 57–61 (1983/1984)Google Scholar
  22. 22.
    Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13, 31–47 (1973)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kollár, J.: Singularities of the Minimal Model Program. With a Collaboration of S. Kovács. Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge (2013)Google Scholar
  24. 24.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. With a Collaboration of C.H. Clemens and A. Corti. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)Google Scholar
  25. 25.
    Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116(1), 133–176 (1982)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex \(3\). Proc. Nat. Acad. Sci. U.S.A. 86(9), 3000–3002 (1989)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mukai, S.: Finite generation of the Nagata invariant rings in A-D-E cases. RIMS1502, Kyoto (2005)Google Scholar
  28. 28.
    Øbro, M.: An algorithm for the classification of smooth Fano polytopes (2007). arXiv:0704.0049
  29. 29.
    Parshin, A.N., Shafarevich, I.R. (eds.): Algebraic Geometry. V. Encyclopaedia of Mathematical Sciences, vol. 47. Springer, Berlin (1999)Google Scholar
  30. 30.
    Reid, M.: The Complete Intersection of Two or More Quadrics. PhD thesis, University of Cambridge (1972)Google Scholar
  31. 31.
    Reid, M.: Decomposition of toric morphisms. In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry, Vol. II. Progress in Mathematics, vol. 36, pp. 395–418. Birkhäuser, Boston (1983)CrossRefGoogle Scholar
  32. 32.
    Szurek, M., Wiśniewski, J.A.: Fano bundles over \({ P}^3\) and \(Q_3\). Pacific J. Math. 141(1), 197–208 (1990)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Voskresenskiĭ, V.E., Klyachko, A.A.: Toric Fano varieties and systems of roots. Izv. Akad. Nauk SSSR Ser. Mat. 48(2), 237–263 (1984) (in Russian)Google Scholar
  34. 34.
    Wiśniewski, J.A.: On Fano manifolds of large index. Manuscripta Math. 70(2), 145–152 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Giulio Codogni
    • 1
  • Andrea Fanelli
    • 2
  • Roberto Svaldi
    • 3
  • Luca Tasin
    • 4
  1. 1.EPFL, SB MATHGEOM CAGLausanneSwitzerland
  2. 2.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany
  3. 3.DPMMS – Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  4. 4.Mathematical Institute of the University of BonnBonnGermany

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