Birational geometry via moduli spaces

TL;DR

This paper explores the birational geometry of Fano threefolds through moduli spaces and Landau--Ginzburg models, proposing new invariants and conjectures to understand rationality and nonrationality.

Contribution

It generalizes Kawamata's categorical approach by incorporating moduli spaces of Landau--Ginzburg models and introduces conjectural invariants related to nonrationality.

Findings

Connections between degenerations of Fano threefolds via projections.

Proposal of new invariants like gaps and phantom categories.

Conjectures on invariants for surfaces of general type and quadric bundles.

Abstract

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.