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Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors

Published:2013-12-04
Printed: Sep 2014
• Kiumars Kaveh,
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
• A. G. Khovanskii,
Department of Mathematics, University of Toronto, Toronto, ON
 Format: LaTeX MathJax PDF

Abstract

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over $\mathbf{k} = \mathbb{C}$. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field $\mathbf{k}$. Secondly we give an isomorphism between the group of Cartier $b$-divisors on the birational class of $X$ and the Grothendieck group of the semigroup of subspaces of rational functions on $X$. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier $b$-divisors and their intersection theory.
 Keywords: intersection number, Cartier divisor, Cartier b-divisor, Grothendieck group
 MSC Classifications: 14C20 - Divisors, linear systems, invertible sheaves 14Exx - Birational geometry

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