$\mathbb{Q}$-Fano threefolds and Laurent inversion

TL;DR

This paper constructs high-codimension non-toric $Q$-Fano threefolds linked to mutation classes of Laurent polynomials, advancing mirror symmetry understanding for these complex algebraic varieties.

Contribution

It introduces new families of non-toric $Q$-Fano threefolds with high codimension, expanding the known examples and establishing the Fano/Landau-Ginzburg correspondence in this context.

Findings

Constructed 54 $Q$-Fano threefolds from mutation classes.

Presented 46 additional $Q$-Fano threefolds with codimensions 10-19.

Established the Fano/Landau-Ginzburg correspondence for these varieties.

Abstract

We construct families of non-toric $\mathbb{Q}$-factorial terminal Fano ($\mathbb{Q}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of mirror symmetry, they are the highest codimension (non-toric) $\mathbb{Q}$-Fano varieties for which we can currently establish the Fano/Landau-Ginzburg correspondence. We construct 46 additional $\mathbb{Q}$-Fano threefolds with codimensions of new examples ranging between 19 and 10. Some of these varieties will be presented as toric complete intersections, and others as Pfaffian varieties.