Zariski's conjecture and Euler-Chow series

Xi Chen; E. javier Elizondo

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arXiv · 1906.08694

Zariski's conjecture and Euler-Chow series

Xi Chen·E. javier Elizondo

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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.

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Xi Chen, E. javier Elizondo. 2019-06-20. Zariski's conjecture and Euler-Chow series. https://arxiv.org/abs/1906.08694

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