Birational geometry via moduli spaces

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 the Hassett-Kuznetsov-Tschinkel program, based on new nonrationality "invariants"-gaps and phantom categories. We formulate several conjectures about these invariants in the case of surfaces of general type and quadric bundles.