The Functor of a Smooth Toric Variety

Abstract

A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In this paper, we generalize this description to the case of maps from Y to an arbitrary smooth toric variety. The data needed to determine such a map consists of a collection of line bundles on Y together with a section of each line bundle. Further, the line bundles must satisfy certain compatbility conditions, and the sections must be nondegenerate in an appropriate sense. In the case of maps from P^m to a smooth toric variety, we get an especially simple description that generalizes the usual way of specifying maps between projective spaces in terms of homogeneous polynomials (of the same degree) that don't vanish simultaneously.