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

Abstract

We construct families of non-toric $\mathbb{Q}$-factorial terminal Fano ($\mathbb{Q}$-Fano) threefolds of codimension $\ge 20$ 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.

arXiv comment40 pages, 6 figures