On some invariants of cubic fourfolds

For a general cubic fourfold X subset of P-5 with Fano variety F, we compute the Hodge numbers of the locus S subset of F of lines of second type and the class of the locus V subset of F of triple lines, using the description of the latter in terms of flag varieties. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.