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!
Best Answer
0) If $P \to X$ is Zariski locally trivial then it is isomorphic to a projectivization of a vector bundle, which is not interesting.
1) The simplest example is the universal conic. Let $X \subset P^5$ be the open subset parameterizing all smooth conics on $P^2$ and $P \subset X\times P^2$ --- the universal conic. Then $P$ over $X$ is a Severi--Brauer variety.
2) Yes. For example you can take for $X'$ a multisection of $P \to X$.
The general reference is the book of Milne "Étale cohomology".
1') Here is a compact example. Let $Y \subset P^5$ be a smooth cubic hypersurface containing a plane $S$. Let $\tilde Y$ be the blowup of $Y$ in $S$. The linear projection from $S$ is a regular map $\tilde Y \to P^2$, the fibers of which are two-dimensional quadrics. Let $D \subset P^2$ be the degeneration divisor (it is a sextic curve) and let $X \to P^2$ be the double covering ramified in $D$ (it is a K3 surface). Let $P \subset Gr(2,6) \times P^2$ be the scheme of lines on fibers of $\tilde Y$ over $P^2$. Then the projection $P \to P^2$ factors through $X$ and the map $P \to X$ is a Severi-Brauer variety.