Geometry of toric varieties

Toric varieties admit a computable description that arise from combinatorialobjects, so-called cones and fans. On the other hand the whole deformation theory ofan isolated singularity is encoded in its semi-universal deformation. More generally,for a complete intersection singularity, deformation is a family over a smooth basespace that is obtained by perturbations of the defining equations. In this thesis, wewant to investigate a description of deformation of ane toric varieties, which wasstudied in Altmann (1995a). It follows that, by the geometric properties of a cone, thesemi-universal deformation, or the total spaces over the components can be describedby completely combinatorial methods. Key points for all our investigations are thegeometric properties of a cone and the notion of a Minkowski summand of somepolyhedron that comes from an ane cross cut of the cone.