Smooth Fano varieties with torus action of Hirzebruch type

We study smooth Fano varieties with torus action by using their description via a specific rational quotient, the so called maximal orbit quotient (MOQ). In the case of torus actions of complexity one, the MOQ turns out to be a projective line, having points as its critical configuration. In this talk we focus on torus actions of complexity two, which turns the MOQ into a surface allowing a bunch of possible quotients. We shortly recall the case where the MOQ is a projective plane with a hyperplane arrangement as its critical configuration. Then, going one step further, we replace the projective plane with a Hirzebruch surface and give first classification results in this setting.