Abstract
AbstractWe give the first examples of $${\mathcal {O}}$$
O
-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over $${\mathbb {P}}^{1}$$
P
1
such that any multi-section has even degree over the base $${\mathbb {P}}^{1}$$
P
1
and show moreover that we can find such a family defined over $${\mathbb {Q}}$$
Q
. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel–Jacobi maps.
Publisher
Springer Science and Business Media LLC
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