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The functor of a smooth toric variety. (English) Zbl 0828.14035
A map \(Y \to \mathbb{P}^n\) is determined by a line bundle \({\mathcal L}\) on \(Y\) and \(n + 1\) global sections of \({\mathcal L}\) which do not vanish simultaneously on \(Y\). 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 consist of a collection of line bundles on \(Y\) together with a section of each line bundle. Further, the line bundles must satisfy certain compatibility conditions, and the section must be nondegenerate in an appropriate sense. These data characterize maps to the toric variety in the strong sense that the toric variety represents the functor assigning to \(Y\) all collections as above. In the case of maps from \(\mathbb{P}^m\) 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 which do not vanish simultaneously.
Reviewer: D.A.Cox (Amherst)

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14E99 Birational geometry
Full Text: DOI
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