Generic fibers of the generalized Springer resolution of type A

It is well known that when the Lie algebra is of type A, D, E the Springer fiber above a subregular nilpotent element is described by the Dynkin diagram and is called the Dynkin curve of the Lie algebra. On the other hand, the closure of the minimal nilpotent orbit is obtained by collapsing the zero section of a cotangent bundle of a projective space Pk. In this article, we are interested in the study of the generalized Springer resolution of type A, we give a complete description of the generalized Springer fiber above a generic singularities showing that it is isomorphic to a Dynkin curve or to a projective space.