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Weak approximation on del Pezzo surfaces of degree 4. (English) Zbl 1211.11077
Poonen, Bjorn (ed.) et al., Arithmetic of higher-dimensional algebraic varieties. Proceedings of the workshop on rational and integral points of higher-dimensional varieties, Palo Alto, CA, USA, December 11–20, 2002. Boston, MA: Birkhäuser (ISBN 0-8176-3259-X/hbk). Progress in Mathematics 226, 235-257 (2004).
Let \(V\) be a del Pezzo surface of degree \(4\) defined over an algebraic number field \(k\). Suppose that \(V(k)\) is not empty. Let \({\mathcal A}\) be the subset of the adelic space \(V({\mathbb A})\) consisting of the points \(\prod P_v\) such that \[ \sum\text{inv}_v(A(P_v))=0\quad\text{in}\;\mathbb Q/\mathbb Z \] for all \(A\) in the Brauer group \(\text{Br}(V)\). P. Salberger and A. N. Skorobogatov have shown that the image of \(V(k)\) is dense in \({\mathcal A}\) [Duke Math. J. 63, No. 2, 517–536 (1991; Zbl 0770.14019)]. The author presents another proof of this theorem.
For the entire collection see [Zbl 1054.11006].

11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
14J25 Special surfaces
14M10 Complete intersections
14G20 Local ground fields in algebraic geometry
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