Discriminants of toric varieties

We study the subvariety of singular sections, the discriminant, of a base point free linear system vertical bar L vertical bar 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.