Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914470702350336 |
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| author | Shah, Saket |
| author_facet | Shah, Saket |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations Shah, Saket Algebraic Geometry Representation Theory 14F08 (Primary) 20G05 (Secondary) 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. |
| title | Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations |
| topic | Algebraic Geometry Representation Theory 14F08 (Primary) 20G05 (Secondary) |
| url | https://arxiv.org/abs/2604.12072 |