Rational curves on Fano threefolds of Picard number one

In this article, we will study the space of very free rational curves on a Fano threefold of Picard number one. We show that a general such rational curve behaves as expected from a deformation point of view. Namely, it is shown that the normal bundle of a general very free rational splits as "evenly" as possible into a direct sum of line bundles. To be more precise, we call a component of rational curves balanced if the normal bundle generically splits as Oa&oplus Ob with |a - b| &le 1. Then all components of very free rational curves on a Fano threefold of Picard number one are balanced with the only exception being the space of conics on P3 .