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Moduli spaces and the inverse Galois problem for cubic surfaces


Authors: Andreas-Stephan Elsenhans and Jörg Jahnel
Journal: Trans. Amer. Math. Soc. 367 (2015), 7837-7861
MSC (2010): Primary 14J15; Secondary 14J20, 14J26, 14G25
Published electronically: March 25, 2015
MathSciNet review: 3391901
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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}$.


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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

DOI: https://doi.org/10.1090/S0002-9947-2015-06277-1
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.
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.