A Verra fourfold is a Fano fourfold which is defined as double cover of $\mathbb{P}^2\times\mathbb{P}^2$ with branch divisor to be $(2,2)$-hypersurface of $\mathbb{P}^2\times\mathbb{P}^2$, which is an index two Fano fourfold of Picard rank two. The $(2,2)$-divisor is a Verra threefold.


But it is known from <https://www.fanography.info/2-6> that there are two types of Verra threefold. I call the $(2,2)$-divisor in $\mathbb{P}^2\times\mathbb{P}^2$ the type I Verra threefold and the one which is double cover of $(1,1)$-divisor of $\mathbb{P}^2\times\mathbb{P}^2$, branched over its anticanonical divisor the type II Verra threefold.


My question is besides the Verra fourfold I defined above, do we have another type of Verra fourfold? It seems that the hyperplane section of the Verra fourfold I defined above is exactly the type II Verra threefold. So I suspect there should be another type of Verra fourfold whose hyperplane section is type I Verra threefold.


Verra threefold is very similar to Gushel-Mukai threefold. We know that there are two types of Gushel-Mukai threefold, and there are also two types of Gushel-Mukai fourfold. The Verra fourfold I defined above is kind of similar to special Gushel-Mukai fourfold, since they are both double cover with branched divisor to be the corresponding type threefold. I was wondering if we have "ordinary" Verra fourfold.



There are only one type of Verra fourfold?
I am not aware of Verra fourfolds of second type, and I think the situation with Verra fourfolds is more similar to the situation with GM sixfolds, where also only one type is available.


