Severi-Brauer variety and finite covering

Question

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!

Answer 1

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.

Answer 2

This is a complement to Sasha's answer, since you asked for an example where $X$ is projective. The simplest projective varieties (curves, rational surfaces) have trivial Brauer group, so there are no non-trivial Severi–Brauer varieties over them. Possibly the simplest interesting example is when $X$ is a K3 surface, and various authors have used elliptic fibrations to construct elements of the Brauer groups of K3 surfaces.

One example is given by Olivier Wittenberg in "Transcendental Brauer–Manin obstruction on a pencil of elliptic curves", Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 259–267, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004 (also on his web site). He constructs an explicit elliptic curve over $\mathbb{Q}(t)$ of which a model is a K3 surface, and finds a non-trivial quaternion algebra lying in the Brauer group of the surface. You could turn this into a bundle of conics over $X$ by the standard construction.

It may well be that there are easier ways to look at this if one's only interested in working over $\mathbb{C}$ – I only know about the situation for varieties defined over $\mathbb{Q}$, where the problem is to find Brauer group elements themselves defined over $\mathbb{Q}$.

I second the recommendation to read the chapter in Milne's book, though it's worth pointing out that the series of three articles on the Brauer group by Grothendieck in "Dix exposés sur la cohomologie des schémas" is actually easier reading that you might anticipate. (And IIRC he explicitly addresses Severi–Brauer varieties.)