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arXiv · 1805.05180

Degree and birationality of multi-graded rational maps

Laurent Busé·Yairon Cid-Ruiz·Carlos D'Andrea
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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. 2018-05-14. Degree and birationality of multi-graded rational maps. https://doi.org/10.1112/plms.12336

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