Abstract
The Cayley–Salmon theorem implies the existence of a 27-sheeted covering space of the parameter space of smooth cubic surfaces, marking each of the 27 lines on each surface. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of \({{\,\mathrm{PGL}\,}}(4,{\mathbb {C}})\). We derive as a consequence of our theorem that over the finite field \({\mathbb {F}}_q\) the average number of lines on a smooth cubic surface equals 1 (away from finitely many characteristics); this average is \(1 + O(q^{-1/2})\) by a standard application of the Weil conjectures.
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Notes
For \(i = 1\) this means the first column is 0. This description of the generators generalizes to \({{\,\mathrm{GL}\,}}(n) \subset M(n)\).
Recall that the fundamental class of an orientable but not necessarily compact n-manifold M without boundary is a generator of \(\overline{H}_{n} (M)\), and the choice of the generator corresponds to the choice of an orientation on M.
References
Allcock, D., Carlson, J.A., Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebr. Geom. 11(4), 659–724 (2002). https://doi.org/10.1090/S1056-3911-02-00314-4. ISSN: 10563911
Brundu, M., Logar, A.: Parametrization of the orbits of cubic surfaces. Transform. Groups 3(3), 209–239 (1998). https://doi.org/10.1007/bf01236873. ISSN: 1083-4362
Bredon, G.E.: Sheaf Theory, Second, ser. Graduate Texts in Mathematics, vol. 170, p. xii+502. Springer, New York (1997). https://doi.org/10.1007/978-1-46120647-7. ISBN: 0-387-94905-4
Bergström, J., Tommasi, O.: The rational cohomology of \({\overline{\cal{M}}}_{4}\). Mathematische Annalen 338(1), 207–239 (2007). https://doi.org/10.1007/s00208-006-0073-z. ISSN: 0025-5831
Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Irreducible Characters. Wiley, Chichester (1993). Wiley Classics Lib, 1985
Cayley, A.: On the triple tangent planes of surfaces of the third order. Camb. Dublin Math. J. 4, 118–132 (1849)
Deligne, P.: Cohomologie étale, ser Lecture Notes in Mathematics, p. iv+312. Springer, Berlin (1977). https://doi.org/10.1007/BFb0091526. Séminaire de géométrie algébrique du Bois-Marie SGA 4\(\frac{1}{2}\), ISBN: 0-387-08066-X
Deligne, P.: La conjecture de weil. II. Inst. Hautes Études. Sci. Publ. Math. 1, 137–252 (1980). ISSN: 0073-8301
Dolgachev, I., van Geemen, B., Kondo, S.: A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces. J. Reine Angew. Math. 588, 99148 (2005). https://doi.org/10.1515/crll.2005.2005.588.99. ISSN: 0075-4102
Harris, J.: Galois groups of enumerative problems. Duke Math. J. 46(4), 685724 (1979). ISSN: 0012-7094
Jordan, C.: Traité des substitutions et des équations algébriques, ser. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux, 1989, pp. xvi\(+\)670, Reprint of the 1870 original, ISBN: 2-87647-021-7 (1989)
Looijenga, E.: Artin groups and the fundamental groups of some moduli spaces. J. Topol. 1(1), 187–216 (2008). https://doi.org/10.1112/jtopol/jtm009. ISSN: 1753-8416
Manin, Y.I.: Cubic forms. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel, vol. 4, 2nd edn, p. x+326. North-Holland Publishing Co., Amsterdam (1986). ISBN: 0-444-87823-8
Milne, J.S.: Lectures on Etale Cohomology (v2.21). , p. 202 (2013)
Naruki, I.: Cross ratio variety as a moduli space of cubic surfaces. Proc. London Math. Soc. (3) 45(1), 1–30 (1982). https://doi.org/10.1112/plms/s3-45.1.1. With an appendix by Eduard Looijenga
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, 1201 (2003). ISSN: 1609-3321
Tommasi, O.: Stable cohomology of spaces of non-singular hypersurfaces. Adv. Math. 265, 428–440 (2014). https://doi.org/10.1016/j.aim.2014.08.005. ISSN: 0001-8708
Totaro, B.: Configuration spaces of algebraic varieties. Topology 35(4), 10571067 (1996). https://doi.org/10.1016/0040-9383(95)00058-5. ISSN: 0040-9383
Vassiliev, V.A.: Complements of discriminants of smooth maps: topology and applications, ser Translations of Mathematical Monographs. Translated from the Russian by B. Goldfarb, vol. 98, p. vi+208. American Mathematical Society, Providence (1992). ISBN: 0-82184555-1
Vassiliev, V.A.: How to calculate the homology of spaces of nonsingular algebraic projective hypersurfaces, Tr. Mat. Inst. Steklova, vol. 225, no. Solitony Geom. Topol. na Perekrest, pp. 132–152. (1999) ISSN: 0371-9685
Acknowledgements
I would like to thank Benson Farb for his invaluable advice and comments throughout the composition of this paper and also for suggesting the problem. I am also grateful to Weiyan Chen, Nir Gadish, Sean Howe and Akhil Mathew for many helpful conversations. I am grateful to Igor Dolgachev for pointing me towards some existing results about moduli spaces of cubic surfaces. I would like to thank Maxime Bergeron, Priyavrat Deshpande, Eduard Looijenga and Jesse Wolfson for their helpful comments to make the paper more readable. Finally, thanks to the anonymous referee for their many comments and suggestions, which helped to greatly improve the paper.
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Das, R. The space of cubic surfaces equipped with a line. Math. Z. 298, 653–670 (2021). https://doi.org/10.1007/s00209-020-02606-5
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DOI: https://doi.org/10.1007/s00209-020-02606-5