  In this paper we connect degenerations of Fano threefolds by projections.
Using Mirror Symmetry we transfer these connections to the side of
Landau--Ginzburg models. Based on that we suggest a generalization of
Kawamata's categorical approach to birational geometry enhancing it via
geometry of moduli spaces of Landau--Ginzburg models. We suggest a conjectural
application to Hasset--Kuznetsov--Tschinkel program based on new nonrationality
"invariants" we consider --- gaps and phantom categories. We make several
conjectures for these invariants in the case of surfaces of general type and
quadric bundles.
