Hi everybody! This is my first post on MO.

Let's work over the field of complex numbers.

Let $f:P\rightarrow X$ be a Severi-Brauer variety over a smooth proper projective algebraic variety $X$ : $f:P\rightarrow X$ is a projective bundle in the complex analytic sense. If I'm correct, to define algebraically such an object locally it is necessary to use étale topology (and the definition is as follows: there exists an \'etale open covering $\{ ( \mu_i: U_i\rightarrow X_i )\}_{i\in I}$ such that $\mu_i^*(P\lvert_{X_i})$ is isomorphic (as schemes) to the trivial bundle $U_i\times \mathbb P^n\rightarrow U_i$ for every $i\in I$.

0) What's going on if one considers Zariski topology intstead of étale topology?

1) What is the simplest example to have in mind (with $X$ projective and smooth) of such a projective bundle that is not trivial (ie. that is not the projectivization of a vector bundle)?

2) Is there always a ramified covering $c:X'\rightarrow X$ such that the pull-back of $P$ by $c$ becomes trivial?

Certainly this is well-known but since I'm not an expert in algebraic geometry, I have been unable to find precise answers to these silly questions in the literature... Any help (answers or references) would be welcome. Thanks!