Regularity of area minimizing currents III: blow-up

We analyze the asymptotic behavior of a 2-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.