dear robert, I am somehow not able to verify your statement. And it is also a bit surprising, now that I thought about it, because it just depends on one parameter. would you mind giving me a hint how you figured this out?

hm okay, I guess that implies that there are no base point free examples...

I would be happier with an example without base points, but thank you for that comment though!

see my edit for the generalization I have in mind.

thank you for this nice answer. I fixed the inaccuracies.

Thank you for this counterexample. Now, I have another question which is related to that.

Okay thanks again. At the moment, I can't see whether my problem reduces to such a situation.