On O'Grady's Generalized Franchetta Conjecture

Abstract

We study relative zero cycles on the universal polarized K3 surface X→Fg of degree 2g−2⁠. It was asked by O'Grady if the restriction of any class in CH2(X) to a closed fiber Xs is a multiple of the Beauville–Voisin canonical class cXs∈CH0(Xs)⁠. Using Mukai models, we give an affirmative answer to this question for g≤10 and g=12,13,16,18,20⁠.