On the Brauer-Manin obstruction for degree four del Pezzo surfaces

2015/03/28 by Jörg Jahnel, Damaris Schindler, Jahnel, Jörg +1
Mathematics · #11G35 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 14F22 #Secondary 14G25 #math.AG #math.NT #msc:11G35 #msc:14F22 #msc:14G25 #msc:14J26

paper · pdf · doi:10.48550/arxiv.1503.08292

arxiv created 2015/03/28 · arxiv updated 2015/03/31

Abstract

We show that, for every integer 1 ≤ d ≤ 4 and every finite set S of places, there exists a degree d del Pezzo surface X over \mathbb Q such that \rm Br(X)/\rm Br(\mathbb Q) ≅ \mathbb Z/2\mathbb Z and the Brauer-Manin obstruction works exactly at the places in S. For d = 4, we prove that in all cases, with the exception of S = \∞\, this surface may be chosen diagonalizably over \mathbb Q.

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