Mirror Symmetry, Laurent Inversion and the Classification of $\mathbb{Q}$-Fano Threefolds

TL;DR

This paper discusses advances in classifying $Q$-Fano threefolds using mirror symmetry, focusing on Laurent inversion techniques that connect Laurent polynomials to Fano varieties.

Contribution

It provides a more conceptual understanding of Laurent inversion, enhancing the construction of Fano varieties from Laurent polynomials within mirror symmetry.

Findings

Improved understanding of Laurent inversion technique.

Enhanced classification framework for $Q$-Fano threefolds.

Better construction methods for Fano varieties from Laurent polynomials.

Abstract

We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual understanding of Laurent inversion, a technique that sometimes allows one to construct a Fano variety $X$ directly from a Laurent polynomial $f$ that corresponds to it under mirror symmetry.