Cubic fourfolds and spaces of rational curves

For a general nonsingular cubic fourfold X subset of p(5) and e > 5 an odd integer, we show that the space Me parametrizing rational curves of degree e on X is non-uniruled. For e > 6 an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of Me is at most one, i.e., passing through a very general point of Me there is at most one rational curve. For e < 5 the spaces Me are fairly well understood and we review what is known.