A remark on the generalized Franchetta conjecture for K3 surfaces
Résumé
A family of K3 surfaces X ---> B has the 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 X(g) ---> F(g) of polarized K3 surfaces of degree 2g-2 has the Franchetta property. It has been proved for small g by Pavic, Shen and Yin, but the general case seems out of reach. We prove here that for all g there is a hypersurface in F(g) such that the corresponding family has the Franchetta property.