A remark on the generalized Franchetta conjecture for K3 surfaces

TL;DR

This paper investigates the Franchetta property for families of polarized K3 surfaces, proving that for all degrees there exists a hypersurface family with this property, extending known results to all genera.

Contribution

The paper demonstrates that for every genus, a hypersurface family of polarized K3 surfaces exhibits the Franchetta property, advancing the understanding of this property beyond previously known cases.

Findings

Existence of hypersurface families with the Franchetta property for all g.

Extension of the Franchetta conjecture to broader classes of K3 families.

Confirmation that the property holds in new geometric contexts.

Abstract

A family of K3 surfaces $\mathscr{X}\rightarrow B$ has the \emph{Franchetta property} if the Chow group of 0-cycles on the generic fiber is cyclic. The generalized Franchetta conjecture proposed by O'Grady asserts that the universal family $\mathscr{X}_g\rightarrow \mathscr{F}_g$ of polarized K3 of degree $2g-2$ has the Franchetta property. While this is known only for small $g$ thanks to \cite{PSY}, we prove that for all $g$ there is a hypersurface in $ \mathscr{F}_g$ such that the corresponding family has the Franchetta property.