Tangent scrolls in prime Fano threefolds

Abstract

In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold X₂₂', contains a 1-dimensional family of intersecting lines. Combined with a result in [Sch] 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.