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Degree and birationality of multi-graded rational maps

Laurent Busé 1 Yairon Cid-Ruiz 2 Carlos d'Andrea 2
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
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.
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Laurent Busé, Yairon Cid-Ruiz, Carlos d'Andrea. Degree and birationality of multi-graded rational maps. Proceedings of the London Mathematical Society, London Mathematical Society, 2020, 121 (4), pp.743-787. ⟨10.1112/plms.12336⟩. ⟨hal-01793578⟩

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