2021/10/26 by Julian Lyczak, Lyczak, Julian, Roman Sarapin +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2110.13687
openalex publication_date 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Várilly-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.