In a recent work Brown and Buczynski described rational maps between toric varieties in terms of their Cox coordinates, lifting a rational map to a multivalued map of Cox rings.
Their construction turns out to be more natural when considering quotient stacks: toric and Mori dream stacks. In a work in progress with Andreas Hochenegger we develop these generalizations.
In the talk I will start with the presentation of an example of Brown-Buczynski's multivalued lifting of a map of toric varieties; for that I will recall some basics in toric geometry.
Then I will explain how this construction becomes natural when looking at quotient stacks. I will briefly recall some notions in stack theory: in particular the notion of canonical stack and the root construction along a divisor over a stack.
Depending on time I will also describe some open question we are working at.