Poles of the topological zeta function associated to an ideal in dimension two

Details Poles of the topological zeta function associated to an ideal in dimension two Poles of the topological zeta function associated to an ideal in dimension two

2007-11-20

arxiv.org/abs/0711.3134v1

Mathematics

Algebraic Geometry

17 pages, to be published in Mathematische Zeitschrift

Scientific paper

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve. In this paper we will study two questions about the poles of this zeta function. First, we will give a criterion to determine whether or not a candidate pole is a pole. It turns out that we can know this immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves. Afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two. The same results are valid for related zeta functions, as for instance the motivic zeta function.

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