Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It seems that there are very few materials in the literature. Is the residue category of $X$ an Enriques category in the sense that its Serre-functor is a involution up to shift by $2$?



Semi-orthogonal decomposition of Verra threefold
The projection of $X$ to each factor is a conic bundle, therefore there is a decomposition
$$
D^b(X) = \langle D^b(\mathbb{P}^2), D^b(\mathbb{P}^2,\mathcal{B}\_0) \rangle,
$$
where $\mathcal{B}\_0$ is the even part of the Clifford algebra. The second component is indeed an Enriques category.


