The Hasse Principle for Lines on del Pezzo Surfaces

In this paper, we consider the following problem: Does there exist a cubic surface over Q which contains no line over Q⁠, yet contains a line over every completion of Q? This question may be interpreted as asking whether the Hilbert scheme of lines on a cubic surface can fail the Hasse principle. We also consider analogous problems, over arbitrary number fields, for other del Pezzo surfaces and complete intersections of two quadrics.