Consider a normal Noetherian scheme $X$. In the case I am most interested in, it's even Cohen-Macaulay. For a locally free coherent sheaf $E$ its projective bundle $\mathbb{P}(E)$ is usually defined as the relative $\mathcal{Proj}(Sym (E))$, where $Sym$ is the symmetric algebra. Sometimes there is also a dual in there, depending on preference and whether you consider a projective space as a space of lines or 1-dimensional quotients.

The same exact definition works for general coherent sheaves. If the stalks are all free, then the fibers of the bundle are also projective spaces, possibly of non-constant dimensions.

Let's say we have a coherent sheaf $E$ on $X$ and $p: \mathbb{P}(E)=Y\to X$. Then if $E$ is locally free of rank $n+1$, then, according to, for example, exercise III.8.4 in Hartshorne, we have

$$p_*(\mathcal{O}_Y(l))=S^l(E),$$ $$R^ip_*(\mathcal{O}_Y(l))=0,\ 0<i<n$$ $$R^np_*(\mathcal{O}_Y(l))=0,\ l>-n-1$$

Since all of this happens primarily because the fibers are projective spaces, can we argue that some of the above holds, for example, that $R^1p_*(\mathcal{O}_Y(l))$ vanishes for $l\geqslant 0$ for a general coherent sheaf $E$, provided that the fibers are indeed projective spaces?

The question Direct image of structure sheaf under blow-up along non-singular subvariety tackles this for the case of a blow-up along a subscheme, but I didn't manage to adapt the proof since I don't have a nice expression for the subscheme $Z\subset Y$ on which $p$ is not an isomorphism.

I also would love to find a reference for this construction of a projective bundle of a coherent sheaf, I very rarely see it treated in this generality.


I just found an MO question Can one prove vanishing of higher direct images fiber-wise?

According to the answer, having $\mathbb{P}^n$ as fibers is not enough when the base scheme $X$ is not normal. Hence I added the normality requirement to the question.



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