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Counting rational points on del Pezzo surfaces with a conic bundle structure. (English) Zbl 1318.14021

Let \(X\) be a del Pezzo surface of degree \(d(X)\) defined over a number field \(k\) and let \(U\) be the Zariski open subset of \(X\) obtained by deleting from \(X\) the finite set of exceptional curves of the first kind. The authors study the number of rational points \[ N(U,k,B):= \text{card}\{x\mid x\in U(k),\;H(x)\leq B\},\quad\text{for }B > 0, \] of bounded anticanonical height \(H\) and conclude as follows. If \(X\subseteq\mathbb{P}^4\), \(d(X)= 4\), and \(X\) contains a non-singular conic defined over \(k\), then \[ N(U,k,B)= O_{\varepsilon, X}(B^{1+ \varepsilon}),\quad\text{for }\varepsilon> 0. \] If \(X: t^2= f(x)\), \(x:=(x_1,x_2,x_3)\), where \(f(x)\in k[x]\), \(f\) is a non-singular form of degree 4, and \(X\) contains a non-singular conic defined over \(k\), then \[ N(U,k,B)= O_{\varepsilon,X}(B^{2+\varepsilon}),\quad\text{for }\varepsilon> 0. \] Moreover, they give a simpler proof of the following theorem of N. Broberg [in: Rational points on algebraic varieties. Basel: Birkhäuser. Prog. Math. 199, 13–35 (2001; Zbl 1080.14517)]. If \(X\subseteq\mathbb{P}^3\), \(d(X)= 4\), and \(X\) contains three coplanar lines defined over \(k\), then \[ N(U,k,B)= O_{\varepsilon,X}(B^{4/3+\varepsilon}),\quad\text{for }\varepsilon> 0. \] These results should be compared with the estimate \[ N(U,k,B)= O_{\varepsilon,X}(B^{1+\varepsilon}),\quad\text{for }\varepsilon> 0 \] expected to hold, according to the Batyrev-Manin conjecture, for any del Pezzo surface.
Reviewer: B. Z. Moroz (Bonn)

MSC:

14G05 Rational points
11G50 Heights
11G35 Varieties over global fields
11D45 Counting solutions of Diophantine equations
14G25 Global ground fields in algebraic geometry
14J26 Rational and ruled surfaces

Citations:

Zbl 1080.14517
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