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Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme. (English) Zbl 1322.11069
The paper under review proves the startling result that the set of cubic surfaces that do not satisfy the Hasse principle are dense in the moduli scheme of cubic surfaces.
An algebraic variety \(V\) (defined, say, over the rational numbers) is said to satisfy the Hasse principle if and only if it contains no rational points, but \(V\) does have real points and points with \(p\)-adic coordinates for every prime \(p\). For example, lines and conic curves are known to satisfy the Hasse principle, but cubic curves do not, in general. Cubic surfaces are also known not to satisfy the Hasse principle, and the paper under review shows that counterexamples are dense in the moduli scheme.
The general strategy is to use the Brauer-Manin obstruction to show the failure of the Hasse principle, in line with a conjecture of Colliot-Thélène [J.-L. Colliot-Thelene and J.-J. Sansuc, Duke Math. J. 48, 421–447 (1981; Zbl 0479.14006)] that the only possible obstructions to the existence of rational points on a cubic surface are the Hasse principle and the Brauer-Manin obstruction. The Brauer-Manin obstruction is calculated using techniques from previous work by the authors [A.-S. Elsenhans and J. Jahnel, Cent. Eur. J. Math. 10, No. 3, 903–926 (2012; Zbl 1276.11112)]).

MSC:
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
14G05 Rational points
14J26 Rational and ruled surfaces
14J10 Families, moduli, classification: algebraic theory
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