Abstract
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L.
Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations
of smooth Fano threefolds by projections was introduced; using Mirror Symmetry
these connections were transferred to the side of Landau--Ginzburg models. In
the paper mentioned above a nice way to connect of Picard rank one Fano
threefolds was found. We apply this approach to all smooth Fano threefolds,
connecting their degenerations by toric basic links. In particular, we find a
lot of Gorenstein toric degenerations of smooth Fano threefolds we need. We
implement mutations in the picture as well. It turns out that appropriate
chosen toric degenerations of the Fanos are given by toric basic links from a
few roots. We interpret the relations we found in terms of Mirror Symmetry.