Arithmetic statistics on cubic surfaces

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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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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Correspondence to 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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