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, 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?<br>
> **Q1.i:**  In general? (With or without assuming that $\pi$ has a section.)<br>
> **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 ($p_g = q = 1$)?<br>
> **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).

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**References**

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


  [1]: http://math.ru.nl/~heckman/Heck_14.pdf
  [2]: http://www.math.hu-berlin.de/~klooster/proefschrift-kloosterman.pdf