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An overview of Manin’s conjecture for del Pezzo surfaces. (English) Zbl 1134.14017

Duke, William (ed.) et al., Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4307-9/pbk). Clay Mathematics Proceedings 7, 39-55 (2007).
Let \(V\) be a projective subvariety of \(\mathbb{P}^n\) defined over \(\mathbb{Q}\) and let \(H\) be the height function on \(\mathbb{P}^n(\mathbb{Q})\) defined by the relation \(H(\vec x):= \max_{0\leq i\leq n}|x_i|\) for \(\vec x\in \mathbb{Z}^{n+1}\), \(\vec x:= (x_0,\dots, x_n)\) with h.c.f. \((x_0,\dots, x_n)= 1\). Given a (Zariski-) open subset \(U\) of \(V\) and a positive real number \(t\), let \[ N(U, t):= \#\{\vec x\mid\vec x\in U(\mathbb{Q}),\,H(\vec x)\leq t\}. \] The author discusses the asymptotic behaviour of the function \(N(U, t)\) for a suitable open subset \(U\) of a del Pezzo surface \(V\) of degree \(d\) as \(t\to\infty\), assuming that \(3\leq d\leq 9\). To illustrate the various ideas and techniques, the author considers the del Pezzo surface \[ V_0: x_0 x_1+ x^2_2= x_1 x_2+ x^2_2+ x_3 x_4= 0 \] in \(\mathbb{P}^3\) and proves the following upper bound: \(N(U_0, t)\ll t(\log t)^5\), where \(U_0\) is the open subset obtained by deleting from \(V_0\) the six lines \[ x_i= x_2= x_j= 0,\;x_0+ x_1= x_1+ x_2= x_j= 0,\;i\in \{0,1\},\;j\in\{3, 4\}. \] Since one can prove that \(N(U_0,t)\gg t(\log t)^5\), the author’s upper bound is of the correct order of magnitude. The author ends his survey article by stating a few open problems.
For the entire collection see [Zbl 1121.11003].
Reviewer: B. Z. Moroz (Bonn)

MSC:

14G05 Rational points
11G35 Varieties over global fields
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14J26 Rational and ruled surfaces
11G25 Varieties over finite and local fields
11D72 Diophantine equations in many variables
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