The derived category of the even Clifford algebra sheaf embeds fully faithfully into that of the orthogonal Grassmannian fibration, enabling a semiorthogonal decomposition for k=2 up to a residual category computed in the smooth case with a conjecture for pencils.
Spinor sheaves on singular quadrics
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Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations
The derived category of the even Clifford algebra sheaf embeds fully faithfully into that of the orthogonal Grassmannian fibration, enabling a semiorthogonal decomposition for k=2 up to a residual category computed in the smooth case with a conjecture for pencils.