Counting points on cubic surfaces, I

Let V be a nonsingular cubic surface defined over Q, let U be the open subset of V obtained by deleting the 27 lines, and denote by N(U, H) the number of rational points in U of height less than H. Manin has conjectured that if V(Q) is not empty then (1) N(U, H) = C1H(log H)(r-1)(1 + o(1)) for some C-1 > 0, where r is the rank of NS(V/Q), the Neron-Severi group of V over Q. In this note we consider the special case when V contains two rational skew lines; and we prove that for some C-2 > 0 and all large enough H, N(U, H) > C2H(log H)(r-1) This is the one-sided estimate corresponding toll). It seems probable that the arguments in this paper could be modified to prove the corresponding result when V contains two skew lines conjugate over Q and each defined over a quadratic extension of Q but we have not attempted to write out the details.