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In this question elliptic surface means a smooth projective complex surface $X$, such that there is an elliptic fibration $\pi \colon X \to C$. (I.e., there is a curve $C$ and a proper map $\pi$, such that almost all fibres are elliptic curves.)

I am aware of 1 and 2. These describe the moduli space of rational elliptic surfaces (unless I stupidly overlooked som parts; I read quite a bit of them, but not every letter). Moreover they assume that the fibration $\pi$ has a section. I would like to know if there is more known about the other cases, in particular those where $C$ is not rational.

First some general questions, asking for literature/references:

Q1: Is there literature on the moduli space of minimal elliptic surfaces?
Q1.i:  In general? (With or without assuming that $\pi$ has a section.)
Q1.ii: In special cases, say when $p_g = q = 1$?

I am particularly interested in whether the Hodge structure of such elliptic surfaces ($p_g = q = 1$) vary when one varies the surface. I want to do this by exhibiting (for every connected component) two elliptic surfaces with different Picard number. However, I have no clue about how the moduli space looks.

Q2.i:  How many components are there in this case ($p_g = q = 1$)?
Q2.ii: What are their dimensions?

Let me finally remark that Remke Kloosterman shows in 2 that there exists extremal elliptic surfaces with these invariants (i.e., maximal Picard number).

References

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  • $\begingroup$ Just a request for clarification: you say " would like to know if there is more known about the other cases, where C is not rational." But the references you refer to only talk about rational surfaces, not all fibrations with rational base. Are you interested in e.g. elliptic K3 surfaces? $\endgroup$
    – user5117
    Commented Feb 2, 2015 at 16:46
  • $\begingroup$ Also, the link in 1 is broken for me because of the lack of www, but the wise and benevolent software won't let me fix it... $\endgroup$
    – user5117
    Commented Feb 2, 2015 at 17:03
  • $\begingroup$ @ArtiePrendergast-Smith — Thanks for the comments. I fixed the link. Concerning the elliptic surfaces I am interested in, I clarified that a bit. So I'm interested in general stuff, but mostly in kodaira-dimension 1, and in particular the case $p_g = q = 1$. $\endgroup$
    – jmc
    Commented Feb 2, 2015 at 17:10

1 Answer 1

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For the case with section and $q=0$ see http://www.math.colostate.edu/~miranda/preprints/weierstrassfibrations.pdf

A similar construction should work in the case (with section, $q$ fixed and $p_g$ sufficiently large) you should get a moduli space together with a morphism to $M_g$.

In the case (with section; $q=p_g=1$) then $p_g$ is ``sufficiently large" and you have that the Weierstrass equation of the elliptic surface depends on the base curve $C$, a choice of a line bundle $L$ of degree 1 on $C$ and two sections in $H^0(L^4)$ and $H^0(L^6)$. You easily can get the dimension of the corresponding moduli space from this. Also this moduli space is obviously connected. Moreover, I believe that the period map is locally an isomorphism in this case, hence different Picard numbers occur.

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  • $\begingroup$ Remke — Thanks for your answer. In the end it was easier then I thought. The only part that I do not really understand is the last sentence. Could you expand a bit on why the period map is locally an isomorphism? $\endgroup$
    – jmc
    Commented Feb 4, 2015 at 5:27
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    $\begingroup$ Consider the period domain for the elliptic surfaces you are interested in. Then the set of points corresponding with elliptic surfaces with maximal Picard number is a countable set. Take now a one-dimensional family of elliptic surfaces, which, as a family of elliptic surfaces, is not isotrivial. Then this yields a curve in the period domain and this curve has to hit a point which corresponds to an elliptic surface with non-maximal Picard number $\endgroup$
    – Remke Kloosterman
    Commented Feb 4, 2015 at 7:17

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