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Arithmetic statistics on cubic surfaces

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Abstract

In this paper, we compute the distributions of various markings on smooth cubic surfaces defined over the finite field \(\mathbb {F}_q\), for example the distribution of pairs of points, ‘tritangents’ or ‘double sixes’. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.

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References

  1. Banwait, B., Fité, F., Loughran, D.: Del pezzo surfaces over finite fields and their frobenius traces. In: Mathematical Proceedings of the Cambridge Philosophical Society (2018). arXiv:1606.00300

  2. Bergvall, O., Gounelas, F.: Cohomology of moduli spaces of del pezzo surfaces (2019). arXiv:1904.10249

  3. Carter, R. W.: Finite groups of Lie type, ser. Wiley Classics Library. Wiley, Chichester. pp. xii+544, Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley–Interscience Publication (1993) ISBN: 0-471-94109-3

  4. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of finite groups. Oxford University Press, Eynsham. pp. xxxiv+252, Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray. (1985) ISBN: 0-19-853199-0

  5. Das, R.: The space of cubic surfaces equipped with a line (2018). arXiv:1803.04146

  6. Das, R.: Cohomology of the universal smooth cubic surface (2019). arXiv:1902.00737

  7. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103(1), 103–161 (1976)

    Article  MathSciNet  Google Scholar 

  8. Frame, J.S.: The classes and representations of the groups of 27 lines and 28 bitangents. Annali di Matematica Pura ed Applicata Serie Quarta 32(1), 83–119 (1951)

    Article  MathSciNet  Google Scholar 

  9. Harris, J.: Galois groups of enumerative problems. Duke Math. J. 46(4), 685–724 (1979)

    Article  MathSciNet  Google Scholar 

  10. Loughran, D., Trepalin, A.: Inverse galois problem for del pezzo surfaces over finite fields (2018). arXiv:1811.06785

  11. Manin, Y.I.: Cubic Forms: Algebra, Geometry, Arithmetic, vol. 4. Elsevier, Amsterdam (1986)

    MATH  Google Scholar 

  12. Naruki, I.: Cross ratio variety as a moduli space of cubic surfaces. Proc. Lond. Math. Soc. 3(1), 1–30 (1982)

    Article  MathSciNet  Google Scholar 

  13. Peters, C.A.M., Steenbrink, J.H.M.: Degeneration of the Leray spectral sequence for certain geometric quotients. Mosc. Math. J. 3(3), 1085–1095 (2003)

    Article  MathSciNet  Google Scholar 

  14. Swinnerton-Dyer, H.P.F.: Cubic surfaces over finite fields. Math. Proc. Camb. Philos. Soc. 149(03), 385–388 (2010)

    Article  MathSciNet  Google Scholar 

  15. Swinnerton-Dyer, H.P.F.: The zeta function of a cubic surface over a finite field. Math. Proc. Camb. Philos. Soc. 63(1), 55–71 (1967)

    Article  MathSciNet  Google Scholar 

  16. Totaro, B.: Configuration spaces of algebraic varieties. Topology 35(4), 1057–1067 (1996)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank Benson Farb for suggesting the project, for guidance and numerous helpful comments throughout the composition of this document. Thanks to Nir Gadish for pointing out an error in an earlier version of Theorem 2.3. I must also thank Nir, and Ben O’Connor, for helping me understand the Grothendieck–Lefschetz trace formula for twisted coefficients. I thank the anonymous referee for their excellent set of detailed and thoughtful comments and suggestions. Thanks to Nate Harman and Nat Mayer for generally helpful conversations. I was supported by the Jump Trading Mathlab Research Fund.

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  1. University of Chicago, Chicago, IL, USA

    Ronno Das

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Das, R. Arithmetic statistics on cubic surfaces. Res Math Sci 7, 23 (2020). https://doi.org/10.1007/s40687-020-00220-9

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