Special Cubic Four-Folds, K3 Surfaces, and the Franchetta Property

Abstract

Abstract O’Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus $g$ is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when $g\leq 10$ and $g=12, 13, 16, 18, 20$), by the work of Pavic–Shen–Yin. In this paper, as a non-commutative analogue, we study the Franchetta property for families of special cubic four-folds (in the sense of Hassett) and relate it to O’Grady’s conjecture for K3 surfaces. Most notably, by using special cubic four-folds of discriminant 26, we prove O’Grady’s generalized Franchetta conjecture for $g=14$, providing the first evidence beyond Mukai models.