Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.


**Q1.** How many such threefolds exist, and how many explicit examples can be given?



Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map
Here is a way to construct such threefolds.


Take $E \to \mathbb P^1$ a non-isotrivial elliptic surface. Take $S$ another surface, say a general type surface. Any rational function on $S$ gives a rational map $f\colon S \to \mathbb P^1$, which may not be well-defined everywhere. However, the graph of $f$ (in generic settings, the blow-up of the indeterminacy locus) does map to $\mathbb P^1$. Call this $S'$. Then $S' \times\_{\mathbb P^1} E$ has an elliptic fibration. It may have singularities, but we can resolve them easily. (If $f$ is a Lefschetz pencil and $E$ is semistable then there should only be node singularities.)


If $S$ is general type, then $S' \times\_{\mathbb P^1} E$ will automatically have Kodaira dimension $2$, and if $S$ is not general type, $S' \times\_{\mathbb P^1} E$ still has Kodaira dimension $2$.


An alternate approach would be to take a sufficiently ample line bundle $L$ on a surface $S$, choose sections $A \in H^0( S, L^4)$ and $B \in H^0( S, L^6)$ and take an elliptic curve in a projective plane bundle over $S$ defined by equations $y^2 = x^3 - A x z^2 - Bz^2 $ for $y$ a section of $L^3$, $x$ a section of $L^2$, and $z$ a section of $\mathcal O\_S$. If $A$ and $B$ are sufficiently generic then the discriminant $-4 A^3 - 27 B^2$ will define a smooth curve in $S$ and this elliptic surface will be smooth, and if $L$ is sufficiently ample then this defines a threefold of Kodaira dimension $2$.


