Big varieties with no nowhere-torsion line bundle

Construct a proper variety that admits a big line bundle but has no nowhere-torsion line bundle.

Background

The paper first gives a proper variety with no nowhere-torsion line bundle, but that example has no big line bundles. It then explains that the more significant question is whether bigness can coexist with the absence of nowhere-torsion line bundles.

The paper subsequently constructs such an example using Ferrand’s pinching construction, so this passage is a motivating question rather than an unresolved problem at the point where it is posed. Under the requested exclusion of questions answered in the same paper, it should not be included as an open problem.