  We study deformations of affine toric varieties. The entire deformation
theory of these singularities is encoded by the so-called versal deformation.
The main goal of our paper is to construct the homogeneous part of some degree
-R of this, i.e. a maximal deformation with prescribed tangent space T^1(-R)
for a given character R. To this aim we use the polyhedron obtained by cutting
the rational cone defining the affine singularity with the hyperplane defined
by [R=1]. Under some length assumptions on the edges of this polyhedron, we
provide the versal deformation for primitive degrees R.
