Rational points of bounded height on Del Pezzo surfaces of degree six (Q1906915)

Let \(V_3\) be a Del Pezzo surface of degree 6 over a number field \(K\). If \(W_3\) is the complement of the union of all exceptional lines of \(V_3\), denote by \(N_{W_3} (-\omega,x)\) the number of \(K\)-rational points of \(W_3\) whose anticanonical height is bounded by \(x\). The author proves the following result. Assume that all the exceptional curves of the first kind of \(V_3\) are defined over \(K\). Then for each number field \(K\) there exists a constant \(c_K\) such that \(N_{W_3} (-\omega,x) \leq c_K x(\log(x))^{3+2r}\), where \(r\) is the rank of the group of units of the ring \({\mathfrak O}_K\) of \(K\). This result is related to a conjecture of Manin.