Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912333390938112 |
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| author | Kuznetsov, Alexander |
| author_facet | Kuznetsov, Alexander |
| contents | Given a flat conic bundle $X/S$ and an abstract spinor bundle $\mathcal{F}$ on $X$ we define a new conic bundle $X_{\mathcal{F}}/S$, called a spinor modification of $X$, such that the even Clifford algebras of $X/S$ and $X_{\mathcal{F}}/S$ are Morita equivalent and the orthogonal complements of $\mathrm{D}^{\mathrm{b}}(S)$ in $\mathrm{D}^{\mathrm{b}}(X)$ and $\mathrm{D}^{\mathrm{b}}(X_{\mathcal{F}})$ are equivalent as well. We demonstrate how the technique of spinor modifications works in the example of conic bundles associated with some nonfactorial 1-nodal prime Fano threefolds. In particular, we construct a categorical absorption of singularities for these Fano threefolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds Kuznetsov, Alexander Algebraic Geometry Given a flat conic bundle $X/S$ and an abstract spinor bundle $\mathcal{F}$ on $X$ we define a new conic bundle $X_{\mathcal{F}}/S$, called a spinor modification of $X$, such that the even Clifford algebras of $X/S$ and $X_{\mathcal{F}}/S$ are Morita equivalent and the orthogonal complements of $\mathrm{D}^{\mathrm{b}}(S)$ in $\mathrm{D}^{\mathrm{b}}(X)$ and $\mathrm{D}^{\mathrm{b}}(X_{\mathcal{F}})$ are equivalent as well. We demonstrate how the technique of spinor modifications works in the example of conic bundles associated with some nonfactorial 1-nodal prime Fano threefolds. In particular, we construct a categorical absorption of singularities for these Fano threefolds. |
| title | Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2502.02082 |