Counterexamples to the Kawamata-Viehweg vanishing on ruled surfaces in positive characteristic (Q627980)

The author obtains counterexamples to the Kawamata-Viehweg vanishing theorem on certain ruled surfaces in positive characteristic. Moreover, he proves that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface \(f:X \to C\) over a smooth curve \(C\), then either all sections of \(f\) are ample or \(C\) is a Tango curve, i.e. the invariant \(n(C)\) of \(C\) introduced by \textit{H. Tango} [Nagoya Math. J. 48, 73--89 (1972; Zbl 0239.14007)] is positive.