  In this paper we prove that any smooth prime Fano threefold, different from
the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting
lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2
(1999), 221-244) this implies that any morphism from a smooth Fano threefold of
index 2 to a smooth Fano threefold of index 1 must be constant, which gives an
answer in dimension 3 to a question stated by Peternell.
