Degree and birationality of multi-graded rational maps. (English) Zbl 1454.13017
Summary: 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.
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 14E05 | Rational and birational maps |
| 13D45 | Local cohomology and commutative rings |
| 13P99 | Computational aspects and applications of commutative rings |
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