On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Firstly, we see that the bases of the miniversal deformations of isolated Q-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤ 2 which are the bases of the miniversal deformations of isolated Q-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension ≥ 3. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension ≥ 3.