Strictly nef divisors and Calabi-Yau threefolds.

A line bundle L is strictly nef if it is ample on every curve. It is classically known that L might not be ample. However a conjecture of Serrano says that K_X + tL is ample for t > dim X + 1.
I discuss this conjecture, indicate a proof in dimension three unless X is Calabi-Yau and study the Calabi-Yau case in more detail.