On O'Grady's generalized Franchetta conjecture

Abstract

We study relative zero cycles on the universal polarized $K3$ surface $X\to {\mathcal{F}}_g$ of degree $2g-2$. It was asked by O'Grady if the restriction of any class in ${\mathrm{CH}}^2(X)$ to a closed fiber $X_s$ is a multiple of the Beauville-Voisin canonical class $c_{X_s}\in {\mathrm{CH}}_0(X_s)$. Using Mukai models, we give an affirmative answer to this question for $g\le 10$ and $g=12,13,16,18,20$.

9 pages