The Hasse principle for lines on del Pezzo surfaces

In this paper, we consider the following problem: Does there exist a cubic surface over $\mathbb{Q}$ which contains no line over $\mathbb{Q}$, yet contains a line over every completion of $\mathbb{Q}$? 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.