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arXiv · 2110.13687

Quartic del Pezzo surfaces with a Brauer group of order 4

Julian Lyczak·Roman Sarapin
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Abstract

We study arithmetic properties of del Pezzo surfaces of degree 4 for which the Brauer group has the largest possible order using different fibrations into curves. We show that if such a surface admits a conic fibration, then it always has a rational point. We also answer a question of V\'arilly-Alvarado and Viray by showing that the Brauer groups these surfaces cannot be vertical with respect to any projection away from a plane. We conclude that the available techniques for proving existence of rational points or even Zariski density do not directly apply if there is no Brauer-Manin obstruction to the Hasse principle. In passing we pick up the first examples of quartic del Pezzo surfaces with a Brauer group of order 4 for which the failure of the Hasse principle is explained by a Brauer-Manin obstruction.

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Julian Lyczak, Roman Sarapin. 2021-10-26. Quartic del Pezzo surfaces with a Brauer group of order 4. https://arxiv.org/abs/2110.13687

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