Suppose I have a projective variety V over a field k. It is given explicitly in terms of homogeneous equations. Moreover, say I have an explicitly given vector bundle E (in terms of a module representing its sheaf of sections) and I would like to study the projectivization P(E) (in the sense of "global Proj").

I would like to do this for particular explicit examples, but it seems inevitable to get into a computational mess. Writing the resulting scheme P(E) as patched from affine opens, I first get dim(V)+1 opens for V, and it seems that for formulating P(E) patched from affines, I now again need to subdivide each such affine open into r (= rank E) opens for the affine patches realizing the fibres....

-> *Question:* I do not know a lot of geometry sadly, but is it maybe true that each such projective bundle is again a projective scheme (over k, 'affine Proj') and there is some quick neat way of rather directly obtaining explicit homogeneous equations for it?

That'd be particularly great since then I could use computer algebra systems for doing basic computations with explicit examples.

Any comments welcome.