Weak approximation for low degree del Pezzo surfaces (Q2913223)

Let \(k\) be an algebraically closed field of characteristic \(0\). Let \(K:=k(C)\) be the function field of a smooth curve \(C\). This paper studies weak approximation for del Pezzo surfaces of low degree, (e.g., \(\leq 4\)) defined over \(K\). For instance, for degree \(1\) del Pezzo surfaces, the following result is obtained.NEWLINENEWLINETheorem. For every smooth degree \(1\) del Pezzo surface \(S\) defined over the function field \(K\) of the curve \(C\), if there exists a model giving a transversal family of degree \(1\) del Pezzo surfaces \({\mathcal{S}}\) over \(C\), then weak approximation holds at each place \(c\in C\).NEWLINENEWLINE For del Pezzo surfaces of degree \(2\), similar statement as above is also proved.NEWLINENEWLINEIn the function field setting, weak approximation means showing the existence of sections with prescribed jet data in a finite number of fibers. For del Pezzo surfaces of degree at least \(4\), weak approximation at all places has been established by \textit{J.-L. Colliot-Thélène} and \textit{Ph. Gille}, Progr. Math. 226, 121--134 (2004; Zbl 1201.11066)]. Also there are some positive results for degree \(3\) and degree \(2\) del Pezzo surfaces. However, the problem is still open for lower degree del Pezzo surfaces, e.g., degree \(1\). This paper fills in that gap.NEWLINENEWLINEThe method used for establishing these results is similar to that of \textit{B. Hassett} and \textit{Y. Tschinkel} [in: Algebraic geometry, Seattle 2005. Proceedings of the 2005 Summer Research Institute, Seattle, WA, USA, July 25--August 12, 2005. Providence, RI: American Mathematical Society (AMS). 937--955 (2009; Zbl 1169.14306)], combined with the author's earlier work [J. Reine Angew. Math. 665, 189--205 (2012; Zbl 1246.14064)].