Note on the Grothendieck group of subspaces of rational functions and Shokurov's Cartier \(b\)-divisors
type: Item
instance of: scholarly article
zbMATH Open document ID: 1325.14020
DOI: 10.4153/CMB-2013-039-6
author: Kiumars Kaveh
author: A. G. Khovanskii
published in: Canadian Mathematical Bulletin
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.2402
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 6359647
zbMATH Keywords: intersection number
zbMATH Keywords: Cartier divisor
zbMATH Keywords: Cartier \(b\)-divisor
zbMATH Keywords: Grothendieck group
MaRDI profile type: Publication
arXiv ID: 1302.2402
title: Note on the Grothendieck group of subspaces of rational functions and Shokurov's Cartier \(b\)-divisors
review text: Let \(X\) be an irreducible variety of dimension \(n\). In the first part of this paper, the authors extend the intersection theory for subspaces of rational functions on a variety \(X\) over \(k=\mathbb{C}\) to an arbitrary algebraically closed field \(k\).NEWLINENEWLINELet \(k(X)\) be the field of rational functions and \(K(X)\) the set of all non-zero finite dimensional vector subspaces of \(k(X)\). Given \(L,M \in K(X)\), define the product \(LM\) to be the subspace spanned by all the product \(fg\), \(f\in L\) and \(g\in M.\) \(K(X)\) is a commutative semigroup with this product. Using the intersection product in the Chow ring of a product of projective spaces, the authors prove two theorems.NEWLINENEWLINE{Theorem 1.} Let \(L_1,\dots, L_n \in K(X)\) and \(Z\subset X\) be a closed subvariety of \(X\) containing the poles of all rational functions from the \(L_i\) and all the points \(x\) at which all functions from some subspace \(L_i\) vanish. Then there is a non-empty Zariski open subset \(U\subset L_1\times\dots\times L_n\) such that for any \((f_1,\dots, f_n)\in U\), the number of solutions NEWLINE\[NEWLINE \{ x\in X\backslash Z, f_1(x)=\dots=f_n(x)=0 \} NEWLINE\]NEWLINE is finite and is independent of the choice of \(Z\) and \((f_1,\dots, f_n)\in U\). The intersection index of the subspaces \(L_i\), denoted by \([L_1,\dots, L_n]\), is the number of solutions \(\{ x\in X\backslash Z, f_1(x)=\dots=f_n(x)=0 \}\).NEWLINENEWLINE{ Theorem 2}. Let \(L_1', L_1'', L_2\dots, L_n \in K(X)\) and \(L_1=L_1'L_1''\). Then NEWLINE\[NEWLINE[L_1,\dots, L_n]= [L_1', L_2,\dots, L_n]+ [L_1'', L_2,\dots, L_n]. NEWLINE\]NEWLINENEWLINENEWLINEIn the second part of the paper, the authors prove that the following two groups are isomorphic.NEWLINENEWLINEA projective birational model is a proper birational map \(\pi\) from a projective variety \(X_{\pi}\) to \(X\). The Riemann-Zariski space space NEWLINE\[NEWLINE{\mathfrak{X}}=\lim_{\overleftarrow{\pi}} X_{\pi},NEWLINE\]NEWLINE where t…
reviewed by: Jing Zhang
description: scientific article; zbMATH DE number 6359647