Degree and birationality of multi‐graded rational maps

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.

Multiplicity of the Saturated Special Fiber Ring of Height Three Gorenstein Ideals

“…Remark 2.5. For any finitely generated bigraded A-module M and any i ≥ 0, c ∈ Z, we have that H i m (M) c has a natural structure as a finitely generated graded S-module (see, e.g., [ Following [5], to study the algebra F(I) it is enough to consider the degree zero part in the R-grading of the bigraded A-module H 1 m (R(I)). Remark 2.6.…”

“…By identifying F(I) ∼ = [R(I)] 0 and F(I) ∼ = H 0 (X, O X ) (see [5]), we obtain the short exact sequence As customary, we approximate the Rees algebra with the symmetric algebra by using the following natural short exact sequence…”

“…Here we provide some applications that follow straightforwardly from the computation of Theorem A and [5]. More specifically, under the assumptions of Setup 2.1, we give an exact formula for the j-multiplicity of I and we study a rational map…”

“…The saturated special fiber ring is an algebra that was in introduced in [5] and that was initially motivated by the interest of studying rational maps with the use of syzygies and blow-up algebras. This algebra encodes valuable information regarding the degree and birationality of rational maps and is also related to the j-multiplicity of ideals (see [5]). Recently, for the case of perfect ideals of height two, its multiplicity was computed in [8].…”

“…From the Buchsbaum-Eisenbud structure theorem [4], we can assume that I has an alternating minimal presentation matrix ϕ whose non-zero entries are all of the same degree D ≥ 1 and that I is minimally generated by the (n − 1) × (n − 1) pfaffians of ϕ. Accordingly, we can assume that I is minimally generated by n homogeneous polynomials of degree δ = 1 2 (n − 1)D. Following [5], the saturated special fiber ring of I is given by the graded k-algebra…”