Note on the Grothendieck group of subspaces of rational functions and Shokurov's Cartier \(b\)-divisors
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 the limit is taken over all the birational models of \(X\). Shokurov defined the group of Cartier \(b-\)divisors as NEWLINE\[NEWLINE\text{CDiv}({\mathfrak{X}})=\lim_{\overrightarrow{\pi}} \text{CDiv}(X_{\pi}),NEWLINE\]NEWLINE where \(\text{CDiv}(X_{\pi})\) denotes the group of Cartier divisors on the variety \(X_{\pi}\) and limit is taken with respect to the pull back maps NEWLINE\[NEWLINE\text{CDiv}(X_{\pi})\rightarrow \text{CDiv}(X_{\pi'})NEWLINE\]NEWLINE when \((X_{\pi'}, \pi')\) dominates \((X_{\pi}, \pi)\).NEWLINENEWLINEThe authors proves that the group of Cartier \(b\)-divisors \(\text{CDiv}({\mathfrak{X}})\) is isomorphic to the Grothendieck group \(G(X)\) of \(K(X)\) and the isomorphism preserves the intersection index.
scientific article; zbMATH DE number 6359647