Should (quasi-) projective schemes be closed under dependent sums?

[felixwellen]

The Segre-Embedding shows that (quasi-) projective schemes are closed under products. Is that also true for dependent sums? I don't know the classical answer but there might be a more or less immediate generalization of the classical argument using Zariski-choice and the Segre-Embedding.

[mnieper]

Given a locally free bundle E -> X of n-dimensional vector spaces over a scheme X, we can define P(E) -> X, the projectivization of the bundle. In general, the dependent sum of projective schemes will live in such a P(E). If X is projective, we may twist E by O(n) for n >> 0 so that it becomes globally generated, i.e. a quotient of a free bundle, say F. We can then embed P(E) into P(F), and P(F), as a product of two projective schemes, is projective by the Segre embedding.

PS I am going to write up foundational material on projective space very soon.

[hmoeneclaey]

I have added some considerations toward this in the notes: 1) Synthetically, it is enough to prove that a dependent sum of projective spaces (i.e. types merely equal to a projective space) over a projective space is a projective scheme. 2) Assuming that $Aut(\mathbb{P}^n) = PGL_{n+1}$ and $H^2(\mathbb{P}^n,\mathbb{A}^\times) = 0$, it is enough to prove the result for projectivisation of locally finite free bundles. I am unsure either of these assumptions are provable.

Does someone know a nice reference for any locally free bundle to be globally generated when twisted enough? I guess it is related to the so-called ample sheaves, of which I know very little.