Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$. Let $E$ be the Albanese variety of $X$, and let $X\to E$ be the Albanese map (having fixed a point). Let $0$ be a closed point of $E$.

Let $F$ be the scheme-theoretic fibre over $0$. Does $F$ have an irreducible

reducedcomponent? That is, does $F$ have an irreducible component of multiplicity one?

Of course, for a general $0$, the fiber $F$ is smooth. I am wondering whether the Albanese map has a multiple fibre or not.

alwayshappens. (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. $\endgroup$3more comments