Poincaré series and zeta function for an irreducible plane curve singularity
نویسنده
چکیده
The Poincaré series of an irreducible plane curve singularity equals the ζ-function of its monodromy, by a result of Campillo, Delgado and GuseinZade. We derive this fact from a formula of Ebeling and Gusein-Zade relating the Poincaré series of a quasi-homogeneous complete intersection singularity to the Saito dual of a product of ζ-functions. Several cases are known where the ζ-function of the monodromy of an isolated hypersurface singularity is related to the Poincaré series of its coordinate ring. The first instance of this phenomenon was observed by Campillo, Delgado and Gusein-Zade [CDG]: for an irreducible plane curve singularity the ζ-function of the monodromy equals its Poincaré series. The proof is by comparing explicit formulas. All attempts have failed at a more direct proof. But maybe this example is misleading. The results of Ebeling and Gusein-Zade [Eb, EGZ] suggest a more indirect connection. For quasi-homogeneous complete intersection singularities they show that the Poincaré series, corrected by an orbit invariant, is related to the Saito dual of a product of ζ-functions. The precise formulation will be given below. The object of this note is to derive the original result of [CDG] from the formula of [EGZ]. The question remains to give a direct proof of their formula and to explain the meaning of the Saito dual. The key observation, stressed by Teissier in [Zar], is that an irreducible plane curve singularity is a (in some sense equisingular) deformation of a monomial curve with the same semigroup, which is moreover a complete intersection. What is needed is a topological argument, which connects the ζ-function of the plane curve singularity with the product of ζ-functions occurring in the formula of [EGZ]. In the first section we give the definition of the monodromy ζ-function and motivate our conventions by recalling the analogy with number theory. Then we state the formula of Ebeling and Gusein-Zade. In the final section we derive the main result.
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