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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
47/2020

Degree and birationality of multi‐graded rational maps

Laurent Busé, Yairon Cid Ruiz and Carlos D'Andrea

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.

Received:
May 3, 2020
Published:
May 6, 2020
MSC Codes:
14E05, 13D02, 13D45, 13P99

Related publications

inJournal
2020 Repository Open Access
Laurent Busé, Yairon Cid-Ruiz and Francesco D'Andrea

Degree and birationality of multi-graded rational maps

In: Proceedings of the London Mathematical Society, 121 (2020) 4, pp. 743-787