Moduli spaces and the inverse Galois problem for cubic surfaces

We study the moduli space M̃ of marked cubic surfaces. By classical work of A. B. Coble, this has a compactification M̃, which is linearly acted upon by the group W(E6). M̃ is given as the intersection of 30 cubics in P9. For the morphism M̃ → P(1, 2, 3, 4, 5) forgetting the marking, followed by Cle...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2015-11, Vol.367 (11), p.7837-7861
Main Authors: ELSENHANS, ANDREAS-STEPHAN, JAHNEL, JÖRG
Format: Article
Language:English
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Summary:We study the moduli space M̃ of marked cubic surfaces. By classical work of A. B. Coble, this has a compactification M̃, which is linearly acted upon by the group W(E6). M̃ is given as the intersection of 30 cubics in P9. For the morphism M̃ → 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 Q.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2015-06277-1