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In each characteristic, there is a canonical homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of sets definable in the theory of algebraically closed fields. We prove that this homomorphism is an isomorphism in characteristic zero. In positive characteristics, we exhibit specific elements in the kernel of the corresponding homomorphism of Grothendieck semirings. The comparison of these two Grothendieck rings in positive characteristics seems to be an open question, related to the difficult problem of cancellativity of the Grothendieck semigroup of varieties.

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... We will need a criterion for isomorphism in the Grothendieck ring. [1] is correct in characteristic zero, but in characteristic p there is a gap in the proof, since separable maps do not necessarily remain separable when restricted to closed subsets. In our case this is guaranteed by the assumption that all of our fibers are reduced points. ...

We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We show that after applying a high enough twist, any maximal component of this locus consists entirely of sections vanishing along a subscheme of minimal degree. In fact, we will give a more refined description of this locus, which will allow us to deduce its limit in the Grothendieck ring of varieties.

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

Kontsevich proposed a Haar measure on k[[t]}, k a field of characteristic zero, with values in a universal ring that is obtained in a simple manner from the varieties over k: it assigns to the ideal tn&[[t]] the value double script L sign-n, where double script L sign stands for the affine line over k. This leads to a measure with the same value ring for schemes over k[[t]]. Denef & Loeser and Batyrev have shown that this measure gives rise to invariants with remarkable properties. These are new even in case the scheme is obtained from a k-variety by base change.

Let K
0(Var
k
) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X
1, ..., X
n
, Y
1, ..., Y
n
into locally closed subvarieties such that X
i
is isomorphic to Y
i
for all i ≤ n), then [X] = [Y] in K
0(Var
k
). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.

Using the weak factorization theorem we give a simple presentation for the value group of the universal Euler characteristic with compact support for varieties of characteristic zero and describe the value group of the universal Euler characteristic of pairs. This gives a new proof for the existence of natural Euler characteristics with values in the Grothendieck group of Chow motives. A generalization of the presentation to the relative setting allows us to define duality and the six operations.