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 *reduced* component? 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.



Surfaces of general type with $q=1$
You can find plenty of examples with multiple Albanese fibres by considering surfaces *isogenous to a product*, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.


For an explicit situation, you can look at Corollary 2.5 of my paper 


*[On surfaces of general type with $p\_g=q=1$ isogenous to a product of curves](https://www.tandfonline.com/doi/abs/10.1080/00927870801948676)*, Communications in Algebra **36** (2008), no. 6, 2023-2053, [arXiv:math/0601063](https://arxiv.org/pdf/math/0601063.pdf).


