The degree of ruled Fano manifolds (Q2761494)

The author presents an example of a function which bounds in each dimension the degree of Fano manifolds for which there exists a morphism onto \(\mathbb{P}^n\) whose fibers are smooth rational curves. The main result is described in the following theorem.NEWLINENEWLINENEWLINELet \(V\) be the projectivisation of a rank 2 vector bundle on \(\mathbb{P}^n\) with even \(c_1\) and let the anticanonical sheaf of \(V\) be ample. Then the following inequality holds \((-K_V)^{n+1} <2\cdot {144\over 143} \cdot(3/4)^{n\over 2}(n+1)^{2n+1}\).