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?
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

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