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I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it being a member of a base-point-free linear system in a nef-Fano fourfold? What, in anything, is known ...