The ruled-surface characterization of Kawamata–Viehweg counterexamples

Let $C$ be a smooth projective curve. A geometrically ruled surface over $C$ is a ruled surface admitting a morphism to $C$ whose fibres are projective lines. A Tango curve is a smooth projective curve with positive Tango invariant $n(C)>0$.

Ruled-surface characterization. There exists a counterexample to the Kawamata–Viehweg vanishing theorem on some geometrically ruled surface over $C$ if and only if $C$ is a Tango curve.

This is proposed as a characterization of counterexamples on geometrically ruled surfaces. The forward implication is supported by the paper's results, while the converse relies on the unresolved problem concerning ample normalized rank-two bundles over curves of genus at least two; the source gives no resolution status.