Zariski's conjecture and Euler-Chow series
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 Poincare series on curves.
math.AG↗