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.

  • $\begingroup$ 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
  • 2
    $\begingroup$ Beautiful...Many thanks! $\endgroup$
    – Pat
    Apr 14 '20 at 9:14

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.