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A power structure over the Grothendieck ring of varieties


Gusein-Zade, Sabir Medgidovich and Luengo Velasco, Ignacio and Melle Hernández, Alejandro (2004) A power structure over the Grothendieck ring of varieties. Mathematical Research Letters, 11 (1). pp. 49-57. ISSN 1073-2780

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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.

Item Type:Article
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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.

Uncontrolled Keywords:Algebraic-Varieties; Spaces; Geometry
Subjects:Sciences > Mathematics > Number theory
ID Code:16623

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