Projecting Fanos in the mirror

TL;DR

This paper extends the connection between toric degenerations of smooth Fano threefolds and Landau-Ginzburg models using Mirror Symmetry, revealing new relations and Gorenstein degenerations.

Contribution

It applies a previously introduced approach to all smooth Fano threefolds, connecting their degenerations via toric basic links and interpreting these in terms of Mirror Symmetry.

Findings

Identified numerous Gorenstein toric degenerations of smooth Fano threefolds.

Connected degenerations of Fanos through toric basic links from a few roots.

Implemented mutations within the toric degeneration framework.

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.