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An introduction to rationally connected fibrations over curves and surfaces

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Abstract

In this expository note we give an overview on some of the main results in the theory of complex rationally connected fibrations and the existence of rational sections. These results have led to the notion of rational simple connectedness.

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References

  1. Araujo, C., Beheshti, R., Castravet, A.-M., Jabbusch, K., Makarova, S., Taylor, L., Viswanathan, N.: Higher Fano manifolds. Preprint, Enrica Mazzon (2021)

  2. Araujo, C., Castravet, A.-M.: Polarized minimal families of rational curves and higher Fano manifolds. Am. J. Math. 134(1), 87–107 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Araujo, C., Castravet, A.-M.: Classification of 2-Fano manifolds with high index. In: Hassett, B., et al. (eds.) A Celebration of Algebraic Geometry. Clay Mathematics Proceedings, vol. 18. American Mathematical Society, Providence (2013)

    Google Scholar 

  4. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkar, C.: Singularities of linear systems and boundedness of Fano varieties. Preprint, arXiv:1609.05543 (2016)

  6. Birkar, C.: Anti-pluricanonical systems on Fano varieties. Ann. Math. (2) 190(2), 345–463 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beheshti, R., Lehmann, B., Riedl, E., Tanimoto, S.: Moduli spaces of rational curves on Fano threefolds. Preprint (2020)

  8. Behrend, K., Manin, Yu.: Stacks of stable maps and Gromov–Witten invariants. Duke Math. J. 85(1), 1–60 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Campana, F.: Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4) 25(5), 539–545 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coskun, I., Starr, J.: Rational curves on smooth cubic hypersurfaces. Int. Math. Res. Not. IMRN 2009(24), 4626–4641 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)

    Book  MATH  Google Scholar 

  12. Debarre, O.: Variétés rationnellement connexes (d’après T. Graber, J. Harris, J. Starr et A. J. de Jong). Number 290, pages Exp. No. 905, ix, 243–266. (2003). Séminaire Bourbaki. Vol. 2001/2002

  13. DeLand, M.: Relatively very free curves and rational simple connectedness. J. Reine Angew. Math. 699, 1–33 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Jong, A.J., He, X., Starr, J.M.: Families of rationally simply connected varieties over surfaces and torsors for semisimple groups. Publ. Math. Inst. Hautes Études Sci. 114, 1–85 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Jong, A.J., Starr, J.M.: Low degree complete intersections are rationally simply connected. Preprint (2006)

  16. de Jong, A.J., Starr, J.M.: A note on Fano manifolds whose second Chern character is positive. Preprint (2006)

  17. de Jong, A.J., Starr, J.: Higher Fano manifolds and rational surfaces. Duke Math. J. 139(1), 173–183 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Druel, S.: Classes de Chern des variétés uniréglées. Math. Ann. 335(4), 917–935 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fanelli, A., Gruson, L., Perrin, N.: Rational curves on \({V}_5\) and rational simple connectedness. Preprint (2019)

  20. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic Geometry—Santa Cruz 1995, volume 62 of Proceedings of Symposia in Pure Mathematics, pp. 45–96. American Mathematical Society, Providence (1997)

  21. Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. 2(90), 542–575 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  22. Graber, T., Harris, J., Mazur, B., Starr, J.: Rational connectivity and sections of families over curves. Ann. Sci. École Norm. Sup. (4) 38(5), 671–692 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gille, P.: Serre’s conjecture II: a survey. In: Garibaldi, S., Sujatha, R., Suresh, V. (eds.) Quadratic Forms, Linear Algebraic Groups, and Cohomology. Developments in Mathematics, vol. 18, pp. 41–56. Springer, New York (2010)

    Chapter  Google Scholar 

  25. Hacon, C.D., Kovács, S.J.: Classification of higher dimensional algebraic varieties. In: Oberwolfach Seminars, vol. 41. Birkhäuser, Basel (2010)

  26. Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8(1), 51–63 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Harris, J., Roth, M., Starr, J.: Rational curves on hypersurfaces of low degree. J. Reine Angew. Math. 571, 73–106 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Harris, J., Starr, J.: Rational curves on hypersurfaces of low degree. II. Compos. Math. 141(1), 35–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Harris, J., Starr, J.: Rational curves on hypersurfaces of low degree. II. Compos. Math. 141(1), 35–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hurwitz, A.: Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39(1), 1–60 (1891)

    Article  MathSciNet  MATH  Google Scholar 

  31. Iskovskih, V.A.: Fano threefolds. I. Izv. Akad. Nauk SSSR Ser. Mat. 41(3), 516–562 (1977)

    MathSciNet  Google Scholar 

  32. Iskovskih, V.A.: Fano threefolds. II. Izv. Akad. Nauk SSSR Ser. Mat. 42(3), 506–549 (1978)

    MathSciNet  Google Scholar 

  33. Kebekus, S.: Families of singular rational curves. J. Algebr. Geom. 11(2), 245–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kebekus, S.: Boundedness results for singular Fano varieties, and applications to Cremona groups [following Birkar and Prokhorov-Shramov]. Astérisque, (422, Séminaire Bourbaki. Vol. 2018/2019. Exposés 1151–1165):253–290 (2020)

  35. Kebekus, S., Kovács, S.J.: Are rational curves determined by tangent vectors? Ann. Inst. Fourier (Grenoble) 54(1), 53–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the (1998) Japanese original

  37. Kollár, J., Miyaoka, Y., Mori, S.: Rationally connected varieties. J. Algebr. Geom. 1(3), 429–448 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Lang, S.: On quasi algebraic closure. Ann. Math. 2(55), 373–390 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lehmann, B., Tanimoto, S.: On the geometry of thin exceptional sets in Manin’s conjecture. Duke Math. J. 166(15), 2815–2869 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lehmann, B., Tanimoto, S.: Geometric Manin’s conjecture and rational curves. Compos. Math. 155(5), 833–862 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lehmann, B., Tanimoto, S.: Rational curves on prime Fano threefolds of index 1. J. Algebr. Geom. 30(1), 151–188 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002)

    Book  MATH  Google Scholar 

  43. Minoccheri, C.: On the arithmetic of weighted complete intersections of low degree. Math. Ann. 377(1–2), 483–509 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. (2) 116(1), 133–176 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mori, S.: Flip theorem and the existence of minimal models for \(3\)-folds. J. Am. Math. Soc. 1(1), 117–253 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  46. Muratore, G.E.: Betti numbers and pseudoeffective cones in 2-Fano varieties. Adv. Geom. 21(4), 505–514 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Nagaoka, T.: On a sufficient condition for a Fano manifold to be covered by rational \(N\)-folds. J. Pure Appl. Algebra 223(11), 4677–4688 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Shimizu, N., Tanimoto, S.: The spaces of rational curves on del Pezzo threefolds of degree one. Eur. J. Math. 8(1), 291–308 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  49. Suzuki, T.: Higher order minimal families of rational curves and Fano manifolds with nef Chern characters. J. Math. Soc. Jpn. 73(3), 949–964 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  50. Takagi, H., Zucconi, F.: Geometries of lines and conics on the quintic del Pezzo 3-fold and its application to varieties of power sums. Mich. Math. J. 61(1), 19–62 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Takagi, H., Zucconi, F.: The moduli space of genus four even spin curves is rational. Adv. Math. 231(5), 2413–2449 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Voisin, C.: Sections rationnelles de fibrations sur les surfaces et conjecture de Serre [d’après de Jong, He et Starr]. Number 348, pages Exp. No. 1038, ix, 317–337. 2012. Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042

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Acknowledgements

The content of this note is based on a Ph.D. course that the author gave at Università Roma Tre between June and July 2022. The author would like to thank Istituto Nazionale di Alta Matematica “Francesco Severi” for the support and Università Roma Tre for the warm and stimulating environment and the referee for very useful comments. The author is currently supported by the ANR Project “FRACASSO” ANR-22-CE40-0009-01.

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  1. Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France

    Andrea Fanelli

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Fanelli, A. An introduction to rationally connected fibrations over curves and surfaces. Rend. Circ. Mat. Palermo, II. Ser 72, 3137–3151 (2023). https://doi.org/10.1007/s12215-023-00910-7

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