Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations

Abstract

Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a Clifford algebra $\mathrm{D}^b(S,\mathcal{C}l_0)$. \par As progress towards a generalization, we show that for a quadric fibration with a fairly minor condition on the rank of the quadric fibers, the category $\mathrm{D}^b(S,\mathcal{C}l_0)$ embeds fully faithfully into the derived category of the relative orthogonal Grassmannian $\mathrm{D}^b(\mathrm{OGr}(k,\mathcal{Q}))$. When $k = 2$, we use this to produce a semiorthogonal decomposition of $\mathrm{D}^b(\mathrm{OGr}(2,\mathcal{Q}))$ up to a residual category; we compute this residual category in the smooth case and produce a conjecture for in the case of a pencil of quadrics with smooth base locus.