Towards homological projective duality for Gr (2, 2n)

Consider a Grassmannian Gr (2, V) for an even-dimensional vector space V. Its derived category of coherent sheaves has a Lefschetz exceptional collection with respect to the Pl\"ucker embedding. We consider a variety X₁ of pairs consisting of a degenerate 2-form on V and a line in its kernel. Note that X₁ is generically a P¹-fibration over the Pfaffian variety of degenerate 2-forms on V. We construct an exceptional collection of coherent sheaves on X₁ such that the subcategory of Dᵇ₂₎₇ (X₁) generated by that collection is conjecturally equivalent to the homologically projectively dual category of the Grassmannian.