Projecting Fanos in the Mirror

A new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced by I. Cheltsov et al. in “Birational geometry via moduli spaces”. Using Mirror Symmetry, these connections were transferred to the side of Landau–Ginzburg models, and a nice way to connect the Picard rank one Fano threefolds was described. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find many Gorenstein toric degenerations of the smooth Fano threefolds we consider. We implement mutations in this framework too. It turns out that appropriately chosen toric degenerations of the Fanos are connected by toric basic links from a few roots. We interpret the relations we found in terms of Mirror Symmetry.