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oai:docta.ucm.es:20.500.14352/501282023-08-25T14:00:14Zcom_20.500.14352_14col_20.500.14352_15

A power structure over the Grothendieck ring of varieties Gusein-Zade, Sabir Medgidovich Luengo Velasco, Ignacio Melle Hernández, Alejandro 511 Algebraic-Varieties Spaces Geometry Teoría de números 1205 Teoría de Números 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. 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. Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T09:39:30Z 2023-06-20T09:39:30Z 2004 journal article https://hdl.handle.net/20.500.14352/50128 1073-2780 eng RFBR–01–01–00739 INTAS–00–259 NWO–RFBR– 047.008.005 BFM2001–1488–C02–01 restricted access application/pdf International Press