Spinor Modifications of Conic Bundles and Derived Categories of 1-Nodal Fano Threefolds

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 $$\mathbf D^{\text{b}}(S)$$ in $$\mathbf D^{\text{b}}(X)$$ and $$\mathbf D^{\text{b}}(X_{\mathcal F})$$ 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.