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On cubic surfaces with a rational line. (English) Zbl 1241.14015

For a smooth cubic surface \(S\) defined over \(\mathbb{Q}\) the 27 lines are acted on by a subgroup \(G\) of the Weil group \(W(E_6)\). It is an open problem whether or not each of the 350 possible conjugacy classes of subgroups is realizable in this way. There are 172 conjugacy classes which fix a line, and the present paper reports on a programme which has successfully found cubic surfaces \(S\) over \(\mathbb{Q}\) realizing each of these cases.
The cases in which the subgroup stabilizes a double-six or a pair of Steiner trihedra have already been covered by the authors [Cent. Eur. J. Math. 8, No. 4, 646–661 (2010; Zbl 1203.14040)] and [Int. J. Number Theory, to appear]. For the remaining 76 subgroups the programme began with a naive search over surfaces of low height, which left only 6 cases to be handled.
For these the procedure begins with the observation that a cubic surface \(S\) which contains a line may be transformed into a del Pezzo surface of degree 4 with a rational point. This allows one to use explicit calculations in fields defined by square roots.

MSC:

14J26 Rational and ruled surfaces
14G25 Global ground fields in algebraic geometry
11G35 Varieties over global fields

Citations:

Zbl 1203.14040
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References:

[1] A.-S. Elsenhans, Good models for cubic surfaces, Preprint.
[2] Elsenhans A.-S., Jahnel J.: Cubic surfaces with a Galois invariant double-six. Central European Journal of Mathematics 8, 646–661 (2010) · Zbl 1203.14040 · doi:10.2478/s11533-010-0036-1
[3] A.-S. Elsenhans and J. Jahnel, Cubic surfaces with a Galois invariant pair of Steiner trihedra, to appear in: International Journal of Number Theory. · Zbl 1233.11073
[4] Kunyavskij B. È., Skorobogatov A. N., Tsfasman M. A.: Del Pezzo surfaces of degree four. Mém. Soc. Math. France 37, 1–113 (1989) · Zbl 0705.14039
[5] MalleG. Matzat B.H.: Inverse Galois theory. Springer, Berlin (1999) · Zbl 0932.12006
[6] Yu. I. Manin, Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974. · Zbl 0277.14014
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