Fano fibrations and DK conjecture for relative Grassmann flips

Given a vector bundle $\mathcal{E}$ on a smooth projective variety $B$, the flag bundle $\mathcal{F}l(1,2,\mathcal{E})$ admits two projective bundle structures over the Grassmann bundles $\mathcal{G}r(1,\mathcal{E})$ and $Gr(2,\mathcal{E})$. The data of a general section of a suitably defined line bundle on $\mathcal{F}l(1,2,\mathcal{E})$ defines two varieties: a cover $X_1$ of $B$ and a fibration $X_2$ on $B$ with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of $X_2$ which consists of a list of exceptional objects and a subcategory equivalent to the derived category of $X_1$. As a byproduct, we obtain a new full exceptional collection for the Fano fourfold of degree $12$ and genus $7$. Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor $\mathcal{F}l(1,2,\mathcal{E})$ is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal-Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.