On some invariants of cubic fourfolds

Abstract

Abstract For a general cubic fourfold $X\subset \mathbb {P}^5$ X ⊂ P 5 with Fano variety F , we compute the Hodge numbers of the locus $S\subset F$ S ⊂ F of lines of second type and the class of the locus $V\subset F$ V ⊂ 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.