Nef extremal rays in the equality case for semistable Jacobian elliptic surfaces

Determine whether, for every semistable Jacobian elliptic surface with δ(π)=χ(O_X), every divisor spanning a nonvertical extremal ray of the nef cone has Iitaka dimension at least one.

Background

For a semistable Jacobian elliptic surface, the paper defines δ(π) from the multiplicities of the singular fibers and proves the bounded cohomology property when δ(π)<χ(O_X).

The boundary case δ(π)=χ(O_X) is not resolved. The relevant issue is whether nonvertical nef extremal rays are generated by divisors with positive Iitaka dimension, a condition that would support the bounded cohomology arguments used elsewhere in the paper.

References

When $\delta(\pi)=\chi(\mathcal O_X)$, it is unknown whether $\kappa(X,D)\ge1$ holds for every divisor $D$ spanning a nonvertical extremal ray of $Nef(X)$, seeProblem~1.3.

— Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones  (2609.00592 - Li, 1 Sep 2026) in Introduction, Remark following Corollary 1.1