Torsion order of smooth projective surfaces

TL;DR

This paper investigates the torsion index of smooth projective surfaces with universally trivial Chow groups, linking it to the torsion in the Néron-Severi group and providing a precise algebraic relationship.

Contribution

It establishes a direct connection between the torsion index of such surfaces and the exponent of torsion in their Néron-Severi group, up to a characteristic-related factor.

Findings

The torsion index equals the exponent of torsion in the Néron-Severi group, up to a power of the characteristic.

Provides a cycle-theoretic decomposition related to the Bloch-Srinivas approach.

Extends understanding of torsion phenomena in algebraic geometry for surfaces.

Abstract

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition \`a la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the exponent of the torsion in the N\'eron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.