On a combinatorial classification of minimal quasi-homogeneous 3-folds with a $SL(2)\times G_m$-action

The classification of minimal smooth projective quasi-homogeneous 3-folds with a $\mathrm{SL}(2)\times\mathbb{G}_m$-action has been known due to results of Mori, Mukai, Nakano, Moser-Jauslin, Kebekus, and Guan. The purpose of my talk is to explain a combinatorial approach to this classification using the Luna-Vust theory of spherical embeddings. This approach describes these minimal 3-folds by means of 2-dimensional fans of colored cones.