## 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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