Geography of Landau-Ginzburg models and threefold syzygies

TL;DR

This paper explores the relationship between toric Landau-Ginzburg models, extremal contractions, and minimal model programs, providing explicit degenerations and computing syzygies for Fano threefolds.

Contribution

It establishes conjectural links between extremal contractions, minimal models, and Landau-Ginzburg models, proving these for smooth toric and Fano varieties in low dimensions.

Findings

Proved conjectures for smooth toric varieties.

Established relations between moduli space and model geography.

Computed elementary syzygies for smooth Fano threefolds.

Abstract

We study the behavior of toric Landau-Ginzburg models under extremal contraction and minimal model program. We also establish a relation between the moduli space of toric Landau-Ginzburg models and the geography of central models. We conjecture that there is a correspondence between extremal contractions and minimal model program on Fano varieties, and degenerations of their associated toric Landau-Ginzburg models written explicitly. We prove the conjectures for smooth toric varieties, as well as general smooth Fano varieties in dimensions 2 and 3. As an application, we compute the elementary syzygies for smooth Fano threefolds.