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How can I find an example of a surface with $q=1$ such that $X\to \mathrm{Alb}(X)$ has no multiple fibres? If I start with one of your examples, I could consider ramified coverings of $X$, but I fear this might increase $q$...
@abx That's a nice example, thank you! But I am wondering whether this always happens. (I think I phrased the question in a confusing manner. My apologies.) Basically, given a surface of general type $X$ with $q=1$, my question is whether we can prove that $X\to Alb(X)$ has a multiple fibre.
I was looking for what @YosemiteStan explained. Basically, from what I understand now, non-big divisors are pull-backs of ample divisors along some quotient map $A\to B$. Can you make this into an answer?
That's amazing. Thank you. I knew this was to much to hope for. However, if the kernel is finite, I presume the answer is positive. Or is that also false?
