On O'Grady's generalized Franchetta conjecture for genus 11 K3 surfaces

TL;DR

Using Mukai's genus $11$ program, the authors establish a birational map between the moduli of genus $11$ K3 data and genus $11$ curves, enabling a reduction of the generalized Franchetta conjecture to the tautological generation of $\operatorname{CH}^2(\overline{M}_{11,1})_\mathbb{Q}$. They prove tautological generation for $\overline{M}_{11,1}$ via a diagonal decomposition and a $G$-invariant analysis on a smooth cover, and then lift this through the Mukai bridge to control $\operatorname{CH}^2(\mathcal{S}_{11}')_\mathbb{Q}$. Spreading out to genus $11$ fibers and examining the corresponding K3 data shows that any codimension-two cycle restricts to a multiple of the Beauville–Voisin class $o_{S_x}$, thus proving the conjecture for genus $11$. This result closes the genus-$11$ gap with a novel strategy and suggests a framework potentially extendable to other genera.

Abstract

O'Grady's generalized Franchetta conjecture asks whether any codimension two cycle on the universal polarized K3 surface restricts to a multiple of the Beauville--Voisin class on a given K3 surface. We apply Mukai's program for genus 11 curves and K3 surfaces, together with a result on the tautological generation of the second Chow group of the moduli space of curves, to give an affirmative answer to this conjecture in genus 11.

Theorems & Definitions (44)

• Conjecture 1.1: Generalized Franchetta Conjecture
• Theorem 1.2
• Theorem 1.3: Mukai's program in genus $11$ Muk
• Theorem 1.4
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof