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](http://www.math.ru.nl/~heckman/Heck_14.pdf) and [2](http://www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf). 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](http://www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf) that there exists extremal elliptic surfaces with these invariants (i.e., maximal Picard number).




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**References**


* [1](http://www.math.ru.nl/~heckman/Heck_14.pdf) Gert Heckman and Eduard Looijenga, *The moduli space of rational elliptic surfaces.* [www.math.ru.nl/~heckman/Heck\_14.pdf](http://www.math.ru.nl/~heckman/Heck_14.pdf)
* [2](http://www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf) Remke N. Kloosterman, *Arithmetic and Moduli
of Elliptic Surfaces.* [www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf](http://www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf)



Is there a description of the moduli space of elliptic surfaces?
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.


