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Mar 31, 2021 at 15:09 comment added user1092847 I would argue that $\mathbb V( \mathcal Q)$ as you've defined it is the correct generalization of Vakil's definition of $\mathbb V(\mathcal F)$ (with duals) to the case where $\mathcal F$ is not locally free (but $X$ is projective). That is: you can interpret your construction as dualizing the sheaf F, in the "correct way."
Mar 31, 2021 at 7:45 history answered Nitin Nitsure CC BY-SA 4.0