Abstract
We study lines on smooth cubic surfaces over the field of p-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are 0, 1, 2, 3, 5, 7, 9, 15 or 27. We show that each of these counts is achieved. Probabilistic aspects are investigated by sampling both p-adic and real cubic surfaces from different distributions and estimating the probability of each count.We link this to recent results on probabilistic enumerative geometry. Some experimental results on the Galois groups attached to p-adic cubic surfaces are also discussed.
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Notes
In his work, Segre actually points out that his statement fails in characteristic 2, but this is not correct; see [7, Section 3.1].
This was done prior to the appearance of McKean’s stronger result [7, Theorem 1.3].
Using a Monte-Carlo method.
The probability that a random surface has a certain number of lines is an integral.
In the tables, we use the notation provided by Magma. In particular, \(F_q\) is the Frobenius group \({\mathbb {F}}_q \rtimes {\mathbb {F}}_q^{\times }\), and \(OD_{2^k}\) is the “other-dihedral” group \(C_{2^{k-1}} \rtimes C_2\) with \(C_2\) acting as \(2^{k-2} + 1\).
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Acknowledgements
We thank Claus Fieker, Tom Fisher, Stevan Gajović, Marta Panizzut, Emre Sertöz and Bernd Sturmfels for valuable discussions. We also thank Avinash Kulkarni, Daniel Loughran and Jean Pierre Serre for their valuable comments on earlier versions of this manuscript. Finally, we thank the referee for suggestions that improved the paper.
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Communicated by Henrik Bachmann.
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El Manssour, R.A., El Maazouz, Y., Kaya, E. et al. Lines on p-adic and real cubic surfaces. Abh. Math. Semin. Univ. Hambg. 93, 149–162 (2023). https://doi.org/10.1007/s12188-023-00269-7
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DOI: https://doi.org/10.1007/s12188-023-00269-7