# zbMATH — the first resource for mathematics

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].

##### MSC:
 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
Full Text: