A remark on the generalized Franchetta conjecture for K3 surfaces

Abstract

A family of K3 surfaces $\mathcal{X}\to B$ 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 ${\mathcal{X}}_g\to {\mathcal{F}}_g$ 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 ${\mathcal{F}}_g$ such that the corresponding family has the Franchetta property.