MINKOWSKI SUMS AND HOMOGENEOUS DEFORMATIONS OF TORIC VARIETIES

We investigate those deformations of affine toric varieties (toric singularities) that arise from embedding them into higher dimensional toric varieties as a relative complete intersection. On the one hand, many examples promise that these so-called toric deformations cover a great deal of the entire deformation theory. On the other hand, they can be described explicitly. Toric deformations are related to decompositions (into a Minkowski sum) of cross cuts of the polyhedral cone defining the toric singularity. Finally, we consider the special case of toric Gorenstein singularities. Many of them turn out to be rigid; for the remaining examples the description of their toric deformations becomes easier than in the general case.