Quartic surfaces, their bitangents and rational points

Let X be a smooth quartic surface not containing lines, defined over number field K. We prove that there are only finitely many bitangents to which This result can interpreted as saying certain surface, having vanishing irregularity, contains rational points. In our proof, we use the geometry of lines double solid associated X. somewhat opposite direction, show on any K, set algebraic points in X(\overeline K) quadratic suitable finite extension K' K is Zariski-dense.