  Given a number field $k$ and a positive integer $d$, in this paper we
consider the following question: does there exist a smooth diagonal surface of
degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every
completion of $k$, yet no line over $k$? We answer the problem using Galois
cohomology, and count the number of counter-examples using a result of
Erd\H{o}s.
