Zariski density of rational points on the bitangent component

Determine whether the rational points on a smooth projective model of the bitangent component of the space of conics associated with a general Fano threefold of genus 2 are Zariski dense.

Background

For a general Fano threefold of index 1 and genus 2, the space of conics has two dominant components: one parametrizing totally tangent conics and another, denoted by MbtM_{\mathrm{bt}}, induced by bitangent lines of the branch sextic. The paper establishes non-density of rational points for the totally tangent component using its large irregularity, but the Abel–Jacobi morphism does not resolve the arithmetic of the bitangent component. Consequently, whether rational points on a smooth projective model of MbtM_{\mathrm{bt}} are Zariski dense remains the unresolved contribution to the genus-2 case and affects whether the geometric exceptional set can be Zariski dense.

References

The arithmetic of the bitangent component $M_{\mathrm{bt}}$ is not determined by the Abel--Jacobi morphism and remains the unresolved part of the genus-$2$ case.

Let $X$ be a general Fano threefold of genus $2$ over a number field $F$, and let $M_{\mathrm{bt}}$ be the component of the space of conics induced by bitangent lines of the branch sextic. Are the rational points on a smooth projective model of $M_{\mathrm{bt}}$ Zariski dense?

— The geometric exceptional set for Fano threefolds  (2610.23024 - Gao, 16 Oct 2026) in Question 2.?, Section “Geometry of the parameter spaces of conics,” subsection “Non-density results”