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

Canadian mathematical bulletin

A. G. Khovanskii¹  Kiumars Kaveh² 

[1] Department of Mathematics, University of Toronto, Toronto, ON;Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

关键词: intersection number;    Cartier divisor;    Cartier b-divisor;    Grothendieck group;   

DOI  :  10.4153/CMB-2013-039-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In a previous paper the authors developed an intersection theory forsubspaces of rational functions on an algebraic variety $X$ over $mathbf{k} = mathbb{C}$. In this short note, we first extend this intersectiontheory 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 groupof the semigroup of subspaces of rational functions on $X$. Theconstructed isomorphism moreover preserves the intersection numbers. This provides an alternative pointof view on Cartier $b$-divisors and their intersection theory.

【 授权许可】

Unknown