Degree and birationality of multi-graded rational maps

TL;DR

This paper develops formulas and bounds for the degree of multi-graded rational maps, introduces a new algebraic tool called the saturated special fiber ring, and extends criteria for birationality to the multi-graded context.

Contribution

It introduces the saturated special fiber ring and extends the Jacobian dual criterion to analyze multi-graded rational maps.

Findings

Derived formulas and bounds for the degree of multi-graded rational maps.

Provided effective criteria for birationality in multi-graded settings.

Described equations of the Rees algebra for certain plane rational maps.

Abstract

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.