Maps of toric varieties in Cox coordinates

TL;DR

This paper develops a unified framework for describing rational maps between toric varieties using Cox coordinates, extending the polynomial approach from projective spaces and introducing formal roots of polynomials.

Contribution

It generalizes the polynomial description of rational maps from projective spaces to arbitrary toric varieties via Cox coordinates, including the use of formal roots.

Findings

Unified description of rational maps in Cox coordinates

Introduction of formal roots of polynomials in toric context

Applicable to arbitrary toric varieties

Abstract

The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric arieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.