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On the cut-and-paste property of algebraic varieties. (English) Zbl 1457.14029

The author studies the property of \(K_0(\operatorname{Var})\), more specifically the cut-and-paste property of varieties. Varieties \(X\) and \(Y\) satisfy cut-and-paste property if there are decompositions into locally closed subvarieties \(X = \bigsqcup_{i=1}^k X_i\) and \(Y = \bigsqcup_{i=1}^k Y_i\) such that \(X_i \cong Y_i\) for all \(i\). In particular cut-and-paste property implies equivalence in \(K_0(\operatorname{Var})\) and birational equivalence.
The author proves that equivalence in \(K_0(\operatorname{Var})\) does not necessarily imply cut-and-paste property. In order to show this he gives explicit examples of of cubic fibrations over the projective line. These cubic fibrations are birationally rigid and hence not birationally equivalent to each other which ensures the failure of cut-and-paste property. Note that the examples presented in the paper are not anomalous or esoteric.
The main part of the paper is the proof of equivalence of these cubic fibrations in \(K_0(\operatorname{Var})\). The author does so indirectly through a chain of equivalences each following from a geometrical construction.

MSC:

14E05 Rational and birational maps
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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