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.
Categories: math.AG