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

TL;DR

This paper introduces a method called spinor modification for conic bundles, establishing derived category equivalences and applying it to 1-nodal Fano threefolds to understand their singularities.

Contribution

It develops the concept of spinor modifications for conic bundles and demonstrates their application to Fano threefolds, linking Clifford algebras and derived categories.

Findings

Spinor modifications preserve Morita equivalence of Clifford algebras.

Derived categories of original and modified bundles are equivalent.

Categorical absorption of singularities is achieved for certain 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.