A power structure over the Grothendieck ring of varieties
A power structure over the Grothendieck ring of varieties
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格洛腾迪克品种环上的权力结构
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发表时间：
2004
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通讯作者：
A. Hernández
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作者：
S. Gusein;I. Luengo;A. Hernández
设R是复拟投射簇的Grothendieck半环（乘法半群），或这些簇的Grothendieck环，或由复仿射线的类L局部化的Grothendieck环。我们在这些（半）环上定义了一个幂结构。这意味着，对于系数[Ai]来自R的幂级数A（t）=1+∑i=1∞[Ai]ti，且对于[M]∈R，定义了一个系数也来自R的级数（A（t））[M]，使得指数函数的所有通常性质都成立。在A（t）=（1−t）−1的特殊情况下，级数（A（t））[M]是由M引入的动机zeta函数。卡普拉诺夫作为一个应用，我们表示的生成函数的希尔伯特计划的点，0维子计划，在一个表面上作为一个指数的表面。
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.