Browning, Tim; Jones, Michael Swarbrick Counting rational points on del Pezzo surfaces with a conic bundle structure. (English) Zbl 1318.14021 Acta Arith. 163, No. 3, 271-298 (2014). 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) Cited in 3 Documents 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 Keywords:rational points; del Pezzo surface; conic bundle surface; Batyrev-Manin conjecture; Thue-Siegel-Roth theorem; anticanonical height Citations:Zbl 1080.14517 PDFBibTeX XMLCite \textit{T. Browning} and \textit{M. S. Jones}, Acta Arith. 163, No. 3, 271--298 (2014; Zbl 1318.14021) Full Text: DOI arXiv