ИСТИНА

An algebraic link in the 3-sphere is the intersection of a germ of complex analytic curve $(C,0)\subset({\mathbb{C}}^2,0)$ with the sphere $S^3_{\varepsilon}$ of small radius centred at the origin. It is known (M.Yamamoto) that the topology of an algebraic link in $S^3$ is determined by its Alexander polynomial in several variables. This follows from the fact that the Alexander polynomial determines the combinatorics of the minimal embedded resolution of the curve. The link of a rational surface singularity $(S,0)$ is a rational homology sphere (a homology sphere: the Poincare sphere - for the $E_8$ surface singularity). An algebraic link in it is the intersection of a germ of complex analytic curve $(C,0)\subset(S,0)$ with the sphere of small radius centred at the origin. It was shown that the Alexander polynomial of an algebraic link in the 3-sphere or in the Poincare sphere coincides with the Poincar series of a filtration on the ring of germs of functions defined by the curve. (Apriori this Poincare series is an analytic invariant.) It was shown that, for a curve in the $E_8$-singularity, the Alexander polynomial does not determine, in general, the combinatorics of the minimal resolution, however it does determine the resolution (and thus the topology of the link) under some explicitly described conditions. In the discussed cases the Alexander polynomial can be expressed as an integral with respect to the Euler characteristic over the space of divisors on the singularity. The coincidence of the Poincare series with the Alexander polynomial is related with the fact that on $({\mathbb{C}}^2,0)$ and on $(S_{E_8},0)$ all divisors are Cartier. One can define a natural generalization of the Alexander polynomial of an algebraic link on other surface singularity (the Weil-Poincare series) as the integral over the space of Weil divisors. The Weil-Poincare series is a power series with rational exponents. One can describe to which extent the Weil-Poincare series determines the topology of the curve for rational double point surface singularities. The talk is based on joint works with A.Campillo and F.Delgado.