On O'Grady's generalized Franchetta conjecture

TL;DR

This paper investigates O'Grady's generalized Franchetta conjecture for polarized K3 surfaces, confirming the conjecture for specific genera using Mukai models.

Contribution

It provides an affirmative answer to O'Grady's question for certain genera, expanding the understanding of zero cycles on K3 surfaces.

Findings

Confirmed the conjecture for g ≤ 10

Confirmed the conjecture for g = 12, 13, 16, 18, 20

Used Mukai models to establish results

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 $g \leq 10$ and $g = 12, 13, 16, 18, 20$.