Counterexamples to the Kawamata–Viehweg vanishing on ruled surfaces in positive characteristic

We give counterexamples to the Kawamata–Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata–Viehweg vanishing theorem on a geometrically ruled surface f:X→C, then either C is a Tango curve or all of sections of f are ample.