Note on the Grothendieck group of subspaces of rational functions and Shokurov's b-divisors

TL;DR

This paper extends an intersection theory for subspaces of rational functions to any algebraically closed field and establishes an isomorphism with b-divisors, preserving intersection numbers, offering a new perspective on their relationship.

Contribution

It generalizes the intersection theory to arbitrary algebraically closed fields and constructs an isomorphism with b-divisors that preserves intersection numbers.

Findings

Extended intersection theory to arbitrary algebraically closed fields.

Established an isomorphism between b-divisors and Grothendieck group of subspaces.

Preserved intersection numbers through the isomorphism.

Abstract

In a previous paper the authors develop an intersection theory for subspaces of rational functions on an algebraic variety X over complex numbers. In this note, we first extend this intersection theory to an arbitrary algebraically closed ground field. Secondly we give an isomorphism between the group of 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 approach to b-divisors and their intersection theory.