  We study the subvariety of singular sections, the discriminant, of a base
point free linear system $|L|$ on a smooth toric variety $X$. On one hand we
describe pairs $(X,L)$ for which the discriminant is of low dimension.
Precisely, we collect some bounds on this dimension and classify those pairs
whose dimension differs the bound less than or equal to two. On the other hand
we study the degree of the discriminant for some relevant families on polarized
toric varieties, describing their minimal values and the region of the ample
cone where this minimal is attained.
