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October 2020 Equations Defining Certain Graphs
Youngsu Kim, Vivek Mukundan
Michigan Math. J. 69(4): 675-710 (October 2020). DOI: 10.1307/mmj/1580439628

Abstract

Consider the rational map ϕ:Pkn1[f0::fn]Pkn defined by homogeneous polynomials f0,,fn of the same degree d in a polynomial ring R=k[x1,,xn] over a field k. Suppose that I=(f0,,fn) is a height two perfect ideal satisfying μ(Ip)dimRp for pSpec(R)V(x1,,xn). We study the equations defining the graph of ϕ whose coordinate ring is the Rees algebra R[It]. We provide new methods to construct these equations using the work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals satisfying the conditions above, our methods lead to explicit equations defining Rees algebras of the ideals in these classes. These classes of examples are interesting in that there are no known methods to compute the defining ideal of the Rees algebra of such ideals. Our new methods also give effective criteria to check that ϕ is birational onto its image.

Citation

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Youngsu Kim. Vivek Mukundan. "Equations Defining Certain Graphs." Michigan Math. J. 69 (4) 675 - 710, October 2020. https://doi.org/10.1307/mmj/1580439628

Information

Received: 22 June 2018; Revised: 30 October 2019; Published: October 2020
First available in Project Euclid: 31 January 2020

MathSciNet: MR4168781
Digital Object Identifier: 10.1307/mmj/1580439628

Subjects:
Primary: 13A30
Secondary: 13D02, 13H15, 14A10, 14E05

Rights: Copyright © 2020 The University of Michigan

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Vol.69 • No. 4 • October 2020
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