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 \(\mathbf D^{\text{b}}(S)\) 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.
