I am considering smooth projective rational threefolds $X$'s with base-point-free anticanonical pencil, i.e., the anticanonical linear system $|-K_X|$ is a base-point-free pencil.
It is a generalization of rational elliptic surfaces. There is one single deformation type for rational elliptic surfaces. However, there are multiple deformation types for $X$'s:
Let $Y_1 = \mathbb P^3$, $Y_2 = \mathbb P^1 \times \mathbb P^1 \times \mathbb P^1 $, $D_i$ be a smooth surface in the linear system $|-K_{Y_i}|$ and $X_i$ be the the blow-up of $Y_i$ along a smooth curve $c_i$ in the linear system $\left | {-K_{Y_i}}|_{D_i} \right |$. Then $X_1$, $X_2$ belong to different deformation types.
One may call $X$ a $K3$-fibered rational threefold because the pencil gives such a fibration.
Starting from toric threefolds, one can build at least thousands of deformation types of such $X$'s.
Is the number of deformation types of such $X$'s bounded?
If so, what would their possible classifications be like? Any references?
