On toric geometry and K-stability of Fano varieties

We present some applications of the deformation theory toric Fano varieties to K-(semi/poly)stability varieties. First, we two examples K-polystable 3-fold with obstructed deformations. In one case, K-moduli spaces and stacks are reducible near closed point associated 3-fold, while in other they non-reduced 3-fold. Second, study K-stability general members families smooth 3-folds by building degenerations 3-folds.