A new spinor modification construction relates conic bundles with equivalent kernel categories, yielding explicit derived category descriptions for 1-nodal Fano threefolds.
Derived Category of certain maximal order on $\mathbb{P}^{2}$
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abstract
We show that the moduli space of $A$-line bundles with minimal second Chern class is a fine moduli space, where $A$ is a maximal quaternion order on $\mathbb{P}^{2}$ ramified along a smooth quartic. We prove that there is a fully faithful embedding from the derived category of this moduli space into the derived category of $A$-modules. Furthermore, we find a semiorthogonal decomposition for $D^{b}(\mathbb{P}^{2},A)$.
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Spinor modifications of conic bundles and derived categories of 1-nodal Fano threefolds
A new spinor modification construction relates conic bundles with equivalent kernel categories, yielding explicit derived category descriptions for 1-nodal Fano threefolds.