The Grothendieck ring of varieties and piecewise isomorphisms
Let \({\mathbb Z}[Var_{k}]\) be the free abelian group having the isomorphism classes of \(k\)-varieties as generators. The additive structure of the Grothendieck ring \(K_{0}(Var_{k})\) is given as the quotient of \({\mathbb Z}[Var_{k}]\) by the relations \([X]=[Z]+[X\backslash {Z}]\) whenever \(X\) is a \(k\)-variety and \(Z\) its closed subvariety. The multiplication on \(K_{0}(Var_{k})\) is given by the formula: \([X] \cdot [Y] :=[(X {\times}_{k} Y)_{red}]\). Not much is known about this ring. It is clear that \([X]=[Y]\) if \(X\) and \(Y\) are piecewise isomorphic, i.e. \(X\) and \(Y\) admit finite partitions into locally closed subvarieties \(X_{1},\dots ,X_{n}\) and \(Y_{1},\dots ,Y_{n}\) respectively such that \(X_{i}\) is isomorphic to \(Y_{i}\) for \(1\leq i \leq n .\) The converse question was asked by \textit{M. Larsen} and \textit{V. Lunts} [Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)]. The authors give an affirmative answer when \(k\) is algebraically closed of characteristic zero for the following cases: \(\dim X\leq 1\) or \(X\) is a smooth projective surface or \(X\) contains finitely many rational curves.
scientific article; zbMATH DE number 5708694
scientific article