Special cubic fourfolds, K3 surfaces and the Franchetta property

TL;DR

This paper explores the Franchetta property for special cubic fourfolds and connects it to O'Grady's conjecture for K3 surfaces, providing new evidence for the conjecture beyond previously known cases.

Contribution

It establishes the Franchetta property for families of special cubic fourfolds of discriminant 26, supporting O'Grady's conjecture for genus 14.

Findings

Proved O'Grady's conjecture for g=14 using special cubic fourfolds.

Connected the Franchetta property of cubic fourfolds to K3 surface conjectures.

Extended understanding of the Franchetta property beyond Mukai models.

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<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fourfolds of discriminant 26, we prove O'Grady's generalized Franchetta conjecture for g=14, providing the first evidence beyond Mukai models.