Deformations of an affine Gorenstein toric pair

We consider deformations of a pair ( X , ∂ X ) , where X is an affine toric Gorenstein variety and ∂ X is its boundary. We compute the tangent and obstruction space for the corresponding deformation functor and for an admissible lattice degree m we construct the miniversal deformation of ( X , ∂ X ) in degrees − k m , for all k ∈ N . This in particular generalizes Altmann's construction of the miniversal deformation of an isolated Gorenstein toric singularity to an arbitrary non-isolated Gorenstein toric singularity. Moreover, we show that the irreducible components of the reduced miniversal deformation are in one to one correspondence with maximal Minkowski decompositions of the polytope P ∩ ( m = 1 ) , where P is the lattice polytope defining X .