# Zariski's conjecture and Euler-Chow series

## TL;DR

This paper explores the connections between Cox ring finite generation, Euler-Chow series rationality, and Zariski's conjecture, proving that finite Cox rings imply rational Poincaré series and establishing rationality for big divisors on surfaces.

## Contribution

It demonstrates that finite generation of Cox rings ensures the rationality of all Poincaré series and extends rationality results to big divisors on surfaces under certain assumptions.

## Key findings

- Finite Cox ring implies all Poincaré series are rational.
- Rationality of multi-variable Poincaré series on curves implies rationality on surfaces.
- Rationality results connect algebraic and geometric properties of varieties.

## Abstract

We study the relations between the finite generation of Cox ring, the rationality of Euler-Chow series and Poincar\'e 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 Poincar\'e series of the variety are rational. We also prove that the multi-variable Poincar\'e series associated to big divisors on a smooth projective surface are rational, assuming the rationality of multi-variable Poincare series on curves.