A power structure over the Grothendieck ring of varieties

Let $\mathcal R$ 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 series $A(t)=1+\sum\limits_{i=1}^\infty [A_i] t^i$ with the coefficients $[A_i]$ from $\mathcal R$ and for $[M]\in {\mathcal R}$, there is defined a series $\left(A(t)\right)^{[M]}$, also with coefficients from $\mathcal R$, so that all the usual properties of the exponential function hold. In the particular case $A(t)=(1-t)^{-1}$, the series $\left(A(t)\right)^{[M]}$ 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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Published 1 January 2004