Publication: A power structure over the Grothendieck ring of varieties
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Publication Date
2004
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International Press
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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 series A(t)=1+∑i=1∞[Ai]ti with the coefficients [Ai] from R and for [M]∈R, there 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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The authors are thankful to Tomás L. Gómez for useful discussions. Partially supported by the grants RFBR–01–01–00739, INTAS–00–259, NWO–RFBR–047.008.005. The last two authors were partially supported by the grant BFM2001–1488–C02–01.
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