Rationality of Euler-Chow series and finite generation of Cox rings

TL;DR

This paper investigates the rationality of Euler-Chow series related to projective varieties and explores the finite generation of Cox rings, providing examples where the series is not rational and proposing conjectures for rationally connected varieties.

Contribution

It presents the first known examples where the Euler-Chow series is not rational and discusses the relationship between Cox ring finite generation and series rationality, including a conjecture for rationally connected varieties.

Findings

The series is not rational for certain blowups of the projective plane.

Examples of varieties with infinitely generated Cox rings where the series is not rational.

Computed Euler-Chow series for Del Pezzo surfaces.

Abstract

In this paper we work with a series whose coefficients are the Euler characteristic of Chow varieties of a given projective variety. For varieties where the Cox ring is defined, it is easy to see that in this case the ring associated to the series is the Cox ring. If this ring is noetherian then the series is rational. It is an open question whether the converse holds. In this paper we give an example showing the converse fails. However we conjecture that it holds when the variety is rationally connected. As an evidence of this conjecture, It is proved that the series is not rational, and in a sense defined, not algebraic, in the case of the blowup of the projective plane at nine or more points in general position. Furthermore, we also treat some other examples of varieties with infinitely generated Cox ring, studied by Mukai and Hassett-Tschinkel. These are the first examples known where the series is not rational. We also compute the series for Del Pezzo surfaces.