 
Summary: Minkowski sums and homogeneous
deformations of toric varieties
Klaus Altmann \Lambda
Abstract
We investigate those deformations of affine toric varieties (toric singular
ities) that arise from embedding them into higher dimensional toric varieties
as a relative complete intersection. On the one hand, many examples promise
that these socalled toric deformations cover a great deal of the entire defor
mation 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.
1 Introduction
(1.1) We want to investigate germs of complex, algebraic singularities Y by
describing their deformation theory. At least for isolated singularities there exists
the socalled (mini) versal deformation which induces all other ones by specializa
tion of parameters. This flat family carries much information about the original
singularity. It is a source for many numerical invariants.
