O'Grady's generalized Franchetta conjecture is concerned with algebraic cycles of codimension 2 on the universal polarized K3 surface of genus g. This conjecture has been confirmed only for certain small values of g. A variant of this conjecture, which replaces Chow groups with Betti cohomology groups, is confirmed for all g by Nicolas Bergeron and Zhiyuan Li. In this talk, I will discuss another variant which replaces Chow groups with Deligne-Beilinson cohomology groups. Besides a sketch of proof, I will mainly focus on two technical points: the fisrt one is to extend the notion of Deligne-Beilinson cohomology to the case of Deligne-Mumford stacks, and the second one is a computation of the numbers of cuspidal and binodal members in certain two-dimensional families of polarized K3 surfaces with ADE singularities. This is joint work with Zhiyuan Li.
