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Kaveh, Kiumars; Khovanskii, A. G.
Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors
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
Categories:14C20, 14Exx

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