A power structure over the Grothendieck ring of varieties

Abstract

Let R be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power seriesthere is defined a series (A(t)) [M ] , also with coefficients from R, so that all the usual properties of the exponential function hold. In the particular case A(t) = (1 − t) −1 , the series (A(t)) [M ] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.

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