Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
Alexander Kuznetsov
TL;DR
The paper develops spinor modifications for flat conic bundles by introducing abstract spinor bundles F that yield a conic bundle X_F/S with a Morita-equivalent Clifford algebra Cliff_0(q_F) to Cliff_0(q) and an equivalence Ker(X/S) ≃ Ker(X_F/S). This framework is applied to nonfactorial 1-nodal Fano threefolds, where explicit spinor bundles F on small resolutions Y → X produce concrete models of Y_F and give decompositions D^b(X) = ⟨P_X, A_X, U_X, O_X⟩ with a universal deformation-absorption object P_X and a Mukai-type component A_X. For the deformation types 12nb, 10na, and 8nb, the orthogonal complements Ker(Y/P^2) correspond to familiar derived categories of a quiver, a genus-2 curve, and a cubic threefold, respectively, while the type 5n case exhibits a Krull–Schmidt partner structure. The results demonstrate a robust categorical absorption of singularities for these Fano threefolds and provide explicit geometric realizations of spinor-modified conic bundles, along with base-change and birational implications of spinor modifications.
Abstract
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.