On Local F-theory Geometries and Intersecting D7-branes

We discuss local F-theory geometries and theirs gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F_0=CP^1 \times CP^1 base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kahler singularities of eight dimensional manifolds considered as sigma model target spaces with eight supercharges. The resulting gauge theory of such local F-theory models are given in terms of Type IIB D7-branes wrapped intersecting F_0. The four dimensional N=1 anomaly cancelation requirement translates into a condition on the associated affine Lie algebras.