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