A remark on the generalized Franchetta conjecture for K3 surfaces

摘要

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 of degree 2(g)-2 has the Franchetta property. While this is known only for small g thanks to 7, we prove that for all g there is a hypersurface in F-g such that the corresponding family has the Franchetta property.