Rationally connectedness and O'Grady's generalized Franchetta conjecture for K3 surfaces

Abstract

O'Grady's generalized Franchetta conjecture asks whether any codimension two cycle on the universal polarized K3 surfaces restricts to a multiple of the Beauville--Voisin class on a given K3 surface. We apply Bergeron--Li's result on the cohomology of universal K3 surfaces to give an affirmative answer to this conjecture when the universal K3 surface is rationally connected, including the case of genus 22.