Fano fibrations and DK conjecture for relative Grassmann flips

Given a vector bundle E on a smooth projective variety B, the flag bundle F l(1,2, E) admits two projective bundle structures over the Grassmann bundles G r(1, E) and G r(2, E). The data of a general section of a suitably defined line bundle on F l(1,2, E) defines two varieties: a cover X₁ of B and a fibration X₂ on B with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of X₂ which consists of a list of exceptional objects and a subcategory equivalent to the derived category of X₁. 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 F l(1, 2, 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.