  For a complete, smooth toric variety Y, we describe the graded vector space
T_Y^1. Furthermore, we show that smooth toric surfaces are unobstructed and
that a smooth toric surface is rigid if and only if it is Fano. For a given
toric surface we then construct homogeneous deformations by means of Minkowski
decompositions of polyhedral subdivisions, compute their images under the
Kodaira-Spencer map, and show that they span T_Y^1.
