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On base change of the fundamental group scheme

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Abstract

We provide for all prime numbers \(p\) examples of smooth projective curves over a field of characteristic \(p\) for which base change of the fundamental group scheme fails. This is intimately related to how \(F\)-trivial vector bundles, i.e. bundles trivialized by a power of the Frobenius morphism, behave in (trivial) families. We conclude with a study of the behavior of \(F\)-triviality in (not necessarily trivial) families.

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Notes

  1. Let \(b\) be in the range \(1\) to \(3pl-3\) then we have \(3pl-3, 3pl - 2, \ldots , 1\) possibilities for \(a\). Taking the sum over these yields \(\sum _{i=1}^{3pl-3} i = \frac{(3pl -3)(3pl - 2)}{2}\).

  2. Actually, we could also work over \(\mathbb {F}_p\) and base change to \(k\) and \(K\).

References

  1. Balaji, V., Parameswaran, A.J.: An analogue of the Narasimhan–Seshadri theorem in higher dimensions and some applications. J. Topol. 4(1), 105–140 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biswas, I, dos Santos, J.P.: Vector bundles trivialized by proper morphisms and the fundamental group scheme, II. arxiv:1004.3609v3 (2011)

  3. Brenner, H.: Computing the tight closure in dimension two. Math. Comput. 74(251), 1495–1518 (2005)

    Article  MATH  Google Scholar 

  4. Brenner, H.: On a problem of Miyaoka. In: van der Geer, G., Moonen, B., Schoof, R. (eds.) Number Fields and Function Fields—Two Parallel Worlds, vol. 239, pp. 51–59. Birkhäuser, Basel (2005)

    Chapter  Google Scholar 

  5. Brenner, H., Stäbler, A.: Dagger closure and solid closure in graded dimension two. Trans. Am. Math. Soc. 365(11), 5883–5910 (2011)

    Article  Google Scholar 

  6. Brenner, H., Stäbler, A.: On the behaviour of strong semistability in geometric deformations. Ill. J, Math. (2014, to appear). arXiv:1107.0877

  7. CoCoATeam: CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it

  8. Deligne, P., Katz, N. (eds.): Groupes de monodromie en géométrie algébrique (=SGA7-II). In: Lecture notes in Mathematics, vol. 340, Springer (1973)

  9. Grothendieck, A., et al.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (=SGA2). North Holland, New York (1968)

    Google Scholar 

  10. Grothendieck, A., et al.: Revêtements étales et groupe fondamental (=SGA1). Springer, Berlin (1970). Also available via arXiv:math/0206203v2

  11. Hartshorne, R.: Ample Subvarieties of Algebraic Varieties. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  12. Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)

    Book  MATH  Google Scholar 

  13. Hartshorne, R.: Deformation Theory. Springer, Berlin (2010)

  14. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Viehweg, Braunschweig (1997)

  15. Kaid, A., Kasprowitz, R.: Semistable vector bundles and Tannaka duality from a computational point of view. Exp. Math. 21(2), 171–188 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lange, H., Stuhler, U.: Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe. Math. Zeitschrift 156, 73–83 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  17. Langton, S.: Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math. 101, 88–110 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mehta, V.B., Subramanian, S.: On the fundamental group scheme. Invent. Math. 148(1), 143–150 (2002)

    Google Scholar 

  19. Mehta, V.B., Subramanian, S.: Some remarks on the local fundamental group scheme. Proc. Indian Acad. Sci. 118(2), 207–211 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton, NJ (1980)

    MATH  Google Scholar 

  21. Miyaoka, Y.: The Chern class and Kodaira dimension of a minimal variety. Algebraic geometry, Sendai 1985. Adv. Stud. Pure Math. 10, 449–476 (1987)

    MathSciNet  Google Scholar 

  22. Nori, M.V.: The fundamental group scheme. Proc. Indian Acad. Sci. (Math. Sci.) 91(2), 73–122 (1982)

  23. Pauly, C.: A smooth counterexample to Nori’s conjecture on the fundamental group scheme. Proc. Am. Math. Soc. 135(9), 2707–2711 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Szamuely, T.: Galois Groups and Fundamental Groups. Cambridge University Press, Cambridge, MA (2009)

    Book  MATH  Google Scholar 

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Acknowledgments

This paper arose from discussions with Holger Brenner in relation to our joint paper [6]. In particular, I thank him for sparking my interest in this problem and for several useful discussions. Furthermore, I thank Manuel Blickle for useful discussions and the referee for a careful reading of an earlier draft and useful comments.

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  1. Johannes Gutenberg-Universität Mainz, Fachbereich 08, Staudingerweg 9, 55099 , Mainz, Germany

    Axel Stäbler

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  1. Axel Stäbler
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Stäbler, A. On base change of the fundamental group scheme. Math. Z. 277, 305–316 (2014). https://doi.org/10.1007/s00209-013-1256-4

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