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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.

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  • $\begingroup$ It may certainly happen. Take an Enriques surface $S$; it admits an elliptic fibration $S\rightarrow \mathbb{P}^1$ with 2 double fibers, say above $0$ and $1$. Now take a double cover $\pi :E\rightarrow \mathbb{P}^1$ branched along 4 points $\neq 0,1$, and let $T$ be the pull back of $S$ by $\pi $; its Albanese map is the projection $p:T\rightarrow E$ with 4 double fibers. Then let $\rho :X\rightarrow T$ be a double covering branched along a smooth, ample curve in $T$, transversal to the double fibers. Its Albanese map $p\circ \rho $ still has 4 double fibers. $\endgroup$
    – abx
    Apr 14 '20 at 7:34
  • $\begingroup$ @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. $\endgroup$
    – Pat
    Apr 14 '20 at 7:57
  • $\begingroup$ No, of course it does not happen always $\endgroup$ Apr 14 '20 at 7:57
  • $\begingroup$ 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 $q$... $\endgroup$
    – Pat
    Apr 14 '20 at 8:00
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    $\begingroup$ Beautiful...Many thanks! $\endgroup$
    – Pat
    Apr 14 '20 at 9:14
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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, Communications in Algebra 36 (2008), no. 6, 2023-2053, arXiv:math/0601063.

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