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@wnx, can you elaborate on the projective bundle part? I think you made a typo, should be $ Gr(3,4) $. If I understand correctly, that has a canonical vector bundle $ \mathcal{E} $ of rank 3 and then taking $ \mathbb{P}( \operatorname{Sym}^3 \mathcal{E}^{*} ) $ gives you a cubic?
Thank you for the answer. I had a question about the assumption of $ D' $ and $ F $ not containing a common component. This seems to rely on the following fact: if the field is infinite, then a finite dimensional vector space is never a finite union of proper subspaces. Or is there some other way to see this?
