Errata to “Discriminants in the Grothendieck ring”

Abstract

The definition ofM in Section 1.1 should be the quotient of K0(VarK) by relations of the form [X] − [Y ] whenever X → Y is a radicial surjective morphism of varieties over K, and all further statements in the paper should use this corrected definition. This quotient of the Grothendieck ring is often taken for applications to motivic integration (see [Mus11, Section 7.2] and [CNS18, Section 4.4]). When K has characteristic 0, these additional relations were already trivial in K0(VarK) (e.g. see [Mus11, Prop 7.25]). The motivic measure of point counting over a finite field still factors through this new definition ofM. This correction is necessary so that the proofs in the paper, in particular those of Theorem 1.13 and in Section 5, are correct. The arguments claim equality inM of [X] and [Y ] where we have a morphism X → Y that is bijective on points over any algebraically closed field. Such an argument is valid in the corrected definition ofM above ([Mus11, Remark A.22]), but is not known to be valid in K0(VarK). We thank Margaret Bilu and Sean Howe for pointing out this mistake and the necessary correction. See [BH19] for further discussion of this issue.