Normal bundles of rational curves on complete intersections

Let X subset of P-n be a general Fano complete intersection of type (d(1), ...,d(k)). If at least one d(i )is greater than 2, we show that X contains rational curves of degree e <= n with balanced normal bundle. If all d(i) are 2 and n >= 2k +1, we show that X contains rational curves of degree e <= n - 1 with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that X is separably rationally connected by exhibiting very free rational curves in X of optimal degrees.