Irrationality of motivic series of Chow varieties

TL;DR

This paper investigates the rationality of the Euler-Chow series associated with Chow varieties, extending the concept to Chow motives, and provides an example demonstrating the series is not rational, raising questions about its geometric significance.

Contribution

The paper generalizes the Euler-Chow series to Chow motives and presents the first example showing the series is not rational.

Findings

The Euler-Chow series can be extended to Chow motives.

An explicit example shows the series is not rational.

Raises questions about the geometric interpretation of the series.

Abstract

The Euler characteristic of all the Chow varieties, of a fixed projective variety, can be collected in a formal power series called the Euler-Chow series. This series coincides with the Hilbert series when the Picard group is a finite generated free abelian group. It is an interesting open problem to find for which varieties this series is rational. A few cases have been computed, and it is suspected that the series is not rational for the blow up of P^2 at nine points in general position. It is very natural to extend this series to Chow motives and ask the question if the series is rational or to find a counterexample. In this short paper we generalized the series and show by an example that the series is not rational. This opens the question of what is the geometrical meaning of the Euler-Chow series.