Degree and birationality of multi‐graded rational maps

Laurent Busé, Yairon Cid Ruiz, and Carlos D'Andrea

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Submission date: 03. May. 2020
published in: Proceedings of the London Mathematical Society, 121 (2020) 4, p. 743-787 
DOI number (of the published article): 10.1112/plms.12336
Bibtex
MSC-Numbers: 14E05, 13D02, 13D45, 13P99
Link to arXiv: See the arXiv entry of this preprint.

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.