What’s not of class TG?

One of the guises that mirror symmetry takes is to try and match deformation classes of Fano varieties with mutation classes of Fano polytopes. One direction of this correspondence is easy: any two polytopes related by mutation define deformation-equivalent toric Fano varieties, while the other requires an existence result about when deformation classes contain a toric variety; a so-called toric degeneration. A Fano variety is of ‘class TG’ (or toric generisation) if its deformation class contains a toric variety.

One of the mirror symmetry conjectures of Coates-Corti-Galkin-Kasprzyk boils down to saying that smooth Fano varieties are of class TG, which seems to be generally believed. The extension of this problem to mildly singular Fano varieties is much muddier. In the case of Fano orbifolds (Fano varieties with at worst finite cyclic quotient singularities), there are several known examples in dimension 2 of orbifold del Pezzo surfaces (some due to me!) which cannot tolerate a toric degeneration and so are not of class TG.

The methods used to approach all the examples that I know of are combinatorial in nature, relying on the properties of Ehrhart series of polytopes that distinguish them from arbitrary power series. It is an interesting and unexplored problem to describe class TG geometrically, and something to which I plan to devote some time.

In the meantime, here is a note explicitly constructing some examples of orbifold del Pezzo surfaces that are not of class TG alongside a discussion of related material.

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