Picard-number-four obstruction in the smooth case

Determine whether there exists a smooth proper toric variety of Picard number $4$ with no tensor-generating line bundle.

Background

The paper proves that smooth proper toric varieties with no tensor-generating line bundle exist in dimension four, and that dimension four is minimal for smooth examples. It also explains that smooth complete toric varieties of Picard number at most three are projective, while the constructed four-dimensional examples have Picard number five. Consequently, Picard number four remains the unresolved minimal possibility.

References

Is there a smooth proper toric variety with Picard number $4$ with no $\otimes$-generating line bundle?

— Tensor-generating line bundles on toric varieties  (2610.22279 - Ito et al., 15 Oct 2026) in Question q:minimal-picard, Section 4 (Existence and non-existence of tensor-generating line bundles)