A computation of invariants of a rational self-map

Let V be a smooth cubic in P and let X = F(V ) be the variety of lines on V . Thus X is a smooth four-dimensional subvariety of the grassmanian G(1, 5), more precisely, the zero locus of a section of SU, where U is the tautological rank-two bundle over G(1, 5). It is immediate from this description that the canonical class of X is trivial. Let F ⊂ V × X be the universal family of lines on V , and let p : F → V , q : F → X be the projections. Beauville and Donagi prove in [BD] that the Abel-Jacobi map AJ : q∗p ∗ : H(V,Z) → H(X,Z) is an isomorphism, at least after tensoring up with Q; since this is also a morphism of Hodge structures, we obtain, by the Noether-Lefschetz theorem for V , that Pic(X) = Z for a sufficiently general X. Lines on cubics have been studied in [CG]. It is shown there that the normal bundle of a general line on a smooth 4-dimensional cubic is OP1 ⊕OP1 ⊕OP1(1) (l is then called ”a line of the first kind”), and that some special lines (”lines of the second kind”) have normal bundle OP1(−1) ⊕ OP1(1) ⊕ OP1(1). The lines of the second kind form a two-parameter subfamily S ⊂ X, and all lines on V are either of the first or of the second kind. An alternative description of lines on a smooth cubic in P from [CG] is as follows: let F (x0, . . . xn) = 0 define a smooth cubic V in P n and consider the Gauss map DV : P n → (P) : x 7→ ( ∂F ∂x0 (x) : . . . : ∂F ∂xn (x))