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Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 3 / 01 September 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 562-572
- Print publication:
- 01 September 2014
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- Article
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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.
