  On a projective surface it is well-known that the set of curves orthogonal to
a nef line bundle is either finite or uncountable. We show that this dichotomy
fails in higher dimension by constructing a nef line bundle on a threefold
which is trivial on countably infinitely many curves. This answers a question
of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with
only a countably infinite set of complete, positive-dimensional subvarieties.
