Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds

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.

v2: 27 pages; to appear in Proc. Steklov Inst. Math