MathDB

O’Grady’s generalized Franchetta conjecture

For every genus g≥2g\ge 2, let π:Xg→Mg\pi:\mathcal{X}_g\to\mathcal{M}_g be the universal family of polarized K3 surfaces of genus gg, and let oS∈CH2(S)o_S\in\mathrm{CH}^2(S) denote the Beauville--Voisin zero-cycle of a fiber S=π−1(t)S=\pi^{-1}(t). Then for every cycle z∈CH2(Xg)z\in\mathrm{CH}^2(\mathcal{X}_g) and every fiber SS of π\pi, there exists λz,S∈Q\lambda_{z,S}\in\mathbb{Q} such that z∣S=λz,SoSz|_S=\lambda_{z,S}o_S in CH2(S)Q\mathrm{CH}^2(S)_{\mathbb{Q}}.

References

Primary source

arXiv

Additional references

algebraic-geometrychow-groupsk3-surfaces
Posted · edited

Progress summary

Refreshed
Claimed progress

A new claimed result covers another important case, but the conjecture remains open for general polarized K3 families.

The conjecture says that every codimension-two cycle on a universal polarized K3 surface restricts on each fiber to a multiple of the Beauville–Voisin zero-cycle. The general assertion remains unresolved.

Known results

  • Pavic, Shen, and Yin (2016): proved the conjecture for g≤10g \le 10 and g=12,13,16,18,20g=12,13,16,18,20.
  • Beauville (2020): proved a weaker statement after restricting to a hypersurface in the moduli space, not the full conjecture.
  • A special-cubic-fourfold construction proves the g=14g=14 case.

September 2026 rational-connectedness case

A September 29, 2026 report describes Yuan Lu’s result that covered codimension-two universal cycles restrict to multiples of the Beauville–Voisin class in rationally connected cases. Lu’s November 21, 2025 paper also claims the previously missing genus-1111 case, using the geometry of genus-1111 curves and tautological Chow classes.

Current status (as of September 2026): The conjecture is claimed for several specific genera and rationally connected cases, while the general polarized K3 moduli problem remains open.

Sources

Solutions 0

No solutions have been posted yet.