abstract: The poset of intersections of a linear hyperplane arrangement is a geometric lattice, and in particular it is shellable. It is natural to ask if the poset of (connected components of) intersections of a toric arrangement is also shellable. This is indeed the case for arrangements associated to root systems, and I will give an idea of how to construct such a shelling (joint work with Emanuele Delucchi and Noriane Girard).