Zariski’s conjecture and Euler–Chow series

Abstract

We study the relations between the finite generation of Cox ring, the rationality of Euler–Chow series and Poincaré series and Zariski’s conjecture on dimensions of linear systems. We prove that if the Cox ring of a smooth projective variety is finitely generated, then all Poincaré series of the variety are rational. We also prove that the multi-variable Poincaré series associated to big divisors on a smooth projective surface are rational, assuming the rationality of multi-variable Poincaré series on curves.