On the first-order deformations of seminormal affine toric varieties

We provide a general description of the module $T_X^1$ of first-order infinitesimal deformations of a not necessarily normal affine toric variety $X$ with a special emphasis on seminormal varieties satisfying Serre’s $(S_2)$ condition. In the surface case we are able to give a very detailed and concrete picture which includes the dimension of each homogeneous component.