Mapping toric varieties into low dimensional spaces

A smooth $d$-dimensional projective variety $X$ can always be embedded into $2d+1$-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any $d$-dimensional projective variety can be mapped injectively to $2d+1$-dimensional projective space. A natural question then arises: what is the minimal $m$ such that a projective variety can be mapped injectively to $m$-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties.