Toric geometry and singularities on K-moduli

An immediate consequence of Kodaira-Akizuki-Nakano vanishing is that smooth Fano varieties have unobstructed deformations. The same holds for singular Fano varieties with mild singularities and small dimension. In this talk I will show how to use the combinatorics of lattice polytopes to construct examples of K-polystable toric Fano varieties with obstructed deformations, dimension at least 3, and canonical singularities. This produces singularities (even reducible / non-reduced / non-Cohen-Macaulay) on K-moduli stacks and K-moduli spaces of Fano varieties (which were recently constructed using K-stability). This is joint work with Anne-Sophie Kaloghiros.