An exponential function on the set of varieties

2002/06/26 by S. M. Gusein‐Zade, S. M. Gusein-Zade, I. Luengo +5
Computer Science · Mathematics · #14A99 #14G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Equations Stability Results #Optimization and Variational Analysis #Polynomial and algebraic computation #math.AG #msc:14A99 #msc:14G10

paper · pdf · doi:10.48550/arxiv.math/0206279

Updated version. Based on a L. Göttsche's result we also show that generating function of the Hilbert scheme of points (0-dimensional subschemes) on a surface is an exponetial of the surface

openalex publication_date 2002/06/26 · arxiv created 2003/01/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cal R be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We introduce a construction which defines operations of taking powers of series over these (semi)rings. This means that, for a power series A(t)=1+∑_i=1^∞ A_i tⁱ with the coefficients A_i from \cal R and for M∈ \cal R, there is defined a series (A(t))^M (with coefficients from \cal R as well) so that all the usual properties of the exponential function hold.We also express in these terms the generating function of the Hilbert scheme of points (0-dimensional subschemes) on a surface.

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