Arithmetic statistics on cubic surfaces
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RESEARCH
Arithmetic statistics on cubic surfaces Ronno Das* * Correspondence:
[email protected] University of Chicago, Chicago, IL, USA
Abstract In this paper, we compute the distributions of various markings on smooth cubic surfaces defined over the finite field Fq , 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.
1 Introduction The classical Cayley–Salmon theorem implies that each smooth cubic surface over an algebraically closed field contains exactly 27 lines (see Sect. 2 for detailed definitions). In contrast, for a surface over a finite field Fq , all 27 lines are defined over Fq but not necessarily over Fq itself. In other words, the action of the Frobenius Frobq permutes the 27 lines and only fixes a (possibly empty) subset of them. It is also classical that the full monodromy group of the 27 lines, i.e. the Galois group of an appropriate extension or cover, is isomorphic to the Weyl group W (E6 ) of type E6 . The Frobenius action on the 27 lines governs much of the arithmetic of the surface S: evidently the pattern of lines defined over Fq and, less obviously, the number of Fq points on S (or UConf n S etc). Work of Bergvall and Gounelas [2] allows us to compute the number of cubic surfaces over Fq where Frobq induces a given permutation, or rather a permutation in a given conjugacy class of W (E6 ). The point-counting results in this paper can be thought of as a combinatorial reinterpretation of their computation. Theorem 1.1 Over the finite field Fq , the number of smooth cubic surfaces on whose 27 lines Frobq acts by a given conjugacy class of W (E6 ) is as in Table 1. The results of Bergvall–Gounelas that we use are cohomological in nature, and we use the Grothendieck–Lefschetz trace formula to convert them to point-counting; see Sect. 2.1. We also directly obtain the rational cohomology of various bundles and covers over the moduli space of smooth cubic surfaces; see Theorem 2.3 and Corollary 2.4. These spaces are the respective moduli spaces of smooth cubic surfaces with various markings of points and lines. The point-counting analogue of this is Proposition 2.6. It is worth noting how Theorem 1.1 relates to the distribution predicted by the Cebotarev density theorem: for a fixed smooth cubic surface defined over Z, the conjugacy class of Frobp acting on the 27 lines of the mod p reduction is distributed (as p → ∞) propor-
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R. Das Res Math Sci (2020)7:23
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Table 1 The number of cubic surfaces over Fq on whose 27 lines Frobq acts by a given conjugacy class of W(E6 ) Conjugacy class c
#{S| Frobq,S ∼ c} #W(E6 ) × # PGL(4, Fq ) #c
(16 )
(q − 2)(q − 3)(q − 5)2
(12 , 22 )
(q + 1)2 (q − 2)(q − 3)
(1−2 , 24 ) (13 , 3) (1−3 , 33 )
(q − 2)(q − 3)(q2 − 2q − 7) q(q + 1)(q2 − q + 1) (q + 1)2 (q2 + q − 3)
2, 3
(32 ) (12 , 2−2 , 42 ) (2, 4)
(q − 2)(q3 − q2 − 2q − 6) (q + 1)3 (q − 2) (q + 1)(
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