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Constraints for the normality of monomial subrings and birationality

Authors: Aron Simis and Rafael H. Villarreal
Journal: Proc. Amer. Math. Soc. 131 (2003), 2043-2048
MSC (2000): Primary 13H10; Secondary 14E05, 14E07, 13B22
Published electronically: November 13, 2002
MathSciNet review: 1963748
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Abstract: Let $k$ be a field and let ${\mathbf F}\subset k[x_1,\ldots,x_{n}]$ be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra $R[\mathbf{F}t]$ and the subring $k[\mathbf{F}]$. If the monomials in $\mathbb{F}$ have the same degree, one of the consequences is a criterion for the $k$-rational map $F\colon{\mathbb P}^{n-1}_k \dasharrow {\mathbb P}^{m-1}_k$ defined by $\mathbf{F}$ to be birational onto its image.

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Additional Information

Aron Simis
Affiliation: Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife, Pe, Brazil

Rafael H. Villarreal
Affiliation: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 México City, D.F., Mexico

Keywords: Birational map, minors, normal ideal, Rees algebras
Received by editor(s): September 10, 2001
Received by editor(s) in revised form: March 7, 2002
Published electronically: November 13, 2002
Additional Notes: The first author was partially supported by a CNPq grant and PRONEX-ALGA (Brazilian Group in Commutative Algebra and Algebraic Geometry)
The second author was supported in part by CONACyT grant 27931E. This author thanks PRONEX-ALGA for their hospitality
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society

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