Moduli spaces and the inverse Galois problem for cubic surfaces
HTML articles powered by AMS MathViewer
- by Andreas-Stephan Elsenhans and Jörg Jahnel PDF
- Trans. Amer. Math. Soc. 367 (2015), 7837-7861 Request permission
Abstract:
We study the moduli space $\widetilde {\mathscr {M}}$ of marked cubic surfaces. By classical work of A. B. Coble, this has a compactification $\widetilde {M}$, which is linearly acted upon by the group $W(E_6)$. $\widetilde {M}$ is given as the intersection of 30 cubics in $\mathbf {P}^9$. For the morphism $\widetilde {\mathscr {M}} \to \mathbf {P}(1,2,3,4,5)$ forgetting the marking, followed by Clebsch’s invariant map, we give explicit formulas, i.e., Clebsch’s invariants are expressed in terms of Coble’s irrational invariants. As an application, we give an affirmative answer to the inverse Galois problem for cubic surfaces over $\mathbb {Q}$.References
- Daniel Allcock and Eberhard Freitag, Cubic surfaces and Borcherds products, Comment. Math. Helv. 77 (2002), no. 2, 270–296. MR 1915042, DOI 10.1007/s00014-002-8340-4
- Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. MR 732439
- A. Cayley, On the triple tangent planes of surfaces of the third order, Cambridge and Dublin Mathematical Journal 4 (1849), 118–132.
- A. Clebsch, Ueber eine Transformation der homogenen Functionen dritter Ordnung mit vier Veränderlichen, J. für die Reine und Angew. Math. 58 (1861), 109–126.
- Arthur B. Coble, Point sets and allied Cremona groups. I, Trans. Amer. Math. Soc. 16 (1915), no. 2, 155–198. MR 1501008, DOI 10.1090/S0002-9947-1915-1501008-7
- Arthur B. Coble, Point sets and allied Cremona groups. III, Trans. Amer. Math. Soc. 18 (1917), no. 3, 331–372. MR 1501073, DOI 10.1090/S0002-9947-1917-1501073-9
- Elisabetta Colombo, Bert van Geemen, and Eduard Looijenga, Del Pezzo moduli via root systems, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, pp. 291–337. MR 2641175, DOI 10.1007/978-0-8176-4745-2_{7}
- Wolfram Decker and David Eisenbud, Sheaf algorithms using the exterior algebra, Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., vol. 8, Springer, Berlin, 2002, pp. 215–249. MR 1949553, DOI 10.1007/978-3-662-04851-1_{9}
- Harm Derksen and Gregor Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130. MR 1918599, DOI 10.1007/978-3-662-04958-7
- Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
- Andreas-Stephan Elsenhans and Jörg Jahnel, The discriminant of a cubic surface, Geom. Dedicata 159 (2012), 29–40. MR 2944518, DOI 10.1007/s10711-011-9643-7
- Andreas-Stephan Elsenhans and Jörg Jahnel, On the arithmetic of the discriminant for cubic surfaces, J. Ramanujan Math. Soc. 27 (2012), no. 3, 355–373. MR 2987232
- Andreas-Stephan Elsenhans and Jörg Jahnel, Cubic surfaces with a Galois invariant double-six, Cent. Eur. J. Math. 8 (2010), no. 4, 646–661. MR 2671217, DOI 10.2478/s11533-010-0036-1
- Andreas-Stephan Elsenhans and Jörg Jahnel, Cubic surfaces with a Galois invariant pair of Steiner trihedra, Int. J. Number Theory 7 (2011), no. 4, 947–970. MR 2812646, DOI 10.1142/S1793042111004253
- Andreas-Stephan Elsenhans and Jörg Jahnel, On cubic surfaces with a rational line, Arch. Math. (Basel) 98 (2012), no. 3, 229–234. MR 2897445, DOI 10.1007/s00013-012-0356-4
- Andreas-Stephan Elsenhans and Jörg Jahnel, On the order three Brauer classes for cubic surfaces, Cent. Eur. J. Math. 10 (2012), no. 3, 903–926. MR 2902222, DOI 10.2478/s11533-012-0042-6
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Bert van Geemen, A linear system on Naruki’s moduli space of marked cubic surfaces, Internat. J. Math. 13 (2002), no. 2, 183–208. MR 1891207, DOI 10.1142/S0129167X0200123X
- A. Grothendieck and J. Dieudonné, Étude cohomologique des faisceaux cohérents (EGA III), Publ. Math. IHES 11 (1961), 17 (1963).
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Bruce Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Springer-Verlag, Berlin, 1996. MR 1438547, DOI 10.1007/BFb0094399
- J. Jahnel, The Brauer-Severi variety associated with a central simple algebra, Linear Algebraic Groups and Related Structures 52 (2000), 1–60.
- Ott-Heinrich Keller, Vorlesungen über algebraische Geometrie, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1974. MR 0429870
- János Kollár, Polynomials with integral coefficients, equivalent to a given polynomial, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 17–27. MR 1445631, DOI 10.1090/S1079-6762-97-00019-X
- A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
- Yu. I. Manin and M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Translated from the Russian by M. Hazewinkel. MR 0460349
- David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
- David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Isao Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982), no. 1, 1–30. With an appendix by Eduard Looijenga. MR 662660, DOI 10.1112/plms/s3-45.1.1
- George Salmon, A treatise on the analytic geometry of three dimensions, Chelsea Publishing Co., New York, 1958. Revised by R. A. P. Rogers; 7th ed. Vol. 1; Edited by C. H. Rowe. MR 0094753
- Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 (French). MR 0103191
- Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. MR 1162313
- C. P. Sousley, Invariants and Covariants of the Cremona Cubic Surface, Amer. J. Math. 39 (1917), no. 2, 135–146. MR 1506315, DOI 10.2307/2370532
- Richard P. Stauduhar, The determination of Galois groups, Math. Comp. 27 (1973), 981–996. MR 327712, DOI 10.1090/S0025-5718-1973-0327712-4
Additional Information
- Andreas-Stephan Elsenhans
- Affiliation: School of Mathematics and Statistics F07, University of Sydney, New South Wales 2006, Sydney, Australia
- Email: stephan@maths.usyd.edu.au
- Jörg Jahnel
- Affiliation: Département Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany
- Email: jahnel@mathematik.uni-siegen.de
- Received by editor(s): March 15, 2013
- Received by editor(s) in revised form: August 11, 2013
- Published electronically: March 25, 2015
- Additional Notes: The first author was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a funded research project.
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 7837-7861
- MSC (2010): Primary 14J15; Secondary 14J20, 14J26, 14G25
- DOI: https://doi.org/10.1090/S0002-9947-2015-06277-1
- MathSciNet review: 3391901