On generalized 7-dimensional Seifert fibrations over complex projective plane

We discuss topological properties of certain $7$-dimensional generalized Seifert fibrations, and, in particular, of the Eschenburg spaces with positive sectional curvature. We compute the values of the first fractional Chern class $\kappa$ of the corresponding $S^3/\mathbb{Z}_q$ bundles over $\mathbb{C} P^1$.