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

Volume 176 / 2016

Jörg Jahnel, Damaris Schindler Acta Arithmetica 176 (2016), 301-319 MSC: Primary 14F22; Secondary 14G25, 14J26, 11G35. DOI: 10.4064/aa8123-8-2016 Published online: 9 November 2016

Abstract

We show that for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree-$d$ del Pezzo surface $X$ over $\mathbb Q$ such that $\operatorname{Br}(X)/\!\operatorname{Br}(\mathbb Q) \cong \mathbb Z/2\mathbb Z$ and the nontrivial Brauer class has a nonconstant local evaluation exactly at the places in $S$. For $d = 4$, we prove that in all cases except $S = \{\infty\}$, this surface may be chosen diagonalisably over $\mathbb Q$.

Authors

  • Jörg JahnelDepartment Mathematik
    Universität Siegen
    Walter-Flex-Str. 3
    D-57068 Siegen, Germany
    e-mail
  • Damaris SchindlerMathematisch Instituut
    Universiteit Utrecht
    Budapestlaan 6
    NL-3584 CD Utrecht, The Netherlands
    e-mail

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