Consider a smooth projective threefold $\overline W$, constructed in section 4 of [this paper](https://arxiv.org/pdf/0810.0957.pdf).
This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb CP^1$ by $(\rho, \psi)$, where $\rho$ is a non-symplectic, non-fixed-point-free involution on the K3 surface and $\psi$ is an involution on $\mathbb CP^1$ (the paper (sec 4) contains more details about the construction).


Then $\overline W$ is simply-connected and it has a smooth K3 surface in its anticanonical system.


Here is my question:



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> Is $\overline W$ irrational? How can one show it?
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Irrationality of some threefolds
I am just posting my comment as an answer. All such threefolds are rational.


By the hypotheses on the involution of the K3 surface, the quotient surface is a rational surface. The projection from the threefold to the rational surface has geometric generic fiber isomorphic to $\mathbb{P}^1$, the other factor in the product. Every fixed point of the involution on $\mathbb{P}^1$ gives a section of this projection.


A normal projective variety that admits a surjective morphism to a target variety is birational (over that target variety) to a product of the target variety and $\mathbb{P}^1$ if and only if the geometric generic fiber is isomorphic to $\mathbb{P}^1$ and there exists a section. Therefore, the threefold is birational to a product of the quotient rational surface and $\mathbb{P}^1$, i.e., it is rational.


