# Brauer-Severi variety

An algebraic variety over a field $k$ that, if considered over the algebraic closure $\overline{k}\;$ of $k$, becomes isomorphic to a projective space.

The arithmetic properties of such varieties were studied in 1932 by F. Severi; F. Châtelet subsequently discovered a connection between Brauer–Severi varieties and central simple algebras (cf. Central simple algebra) over $k$ and the Brauer group.

The simplest non-trivial example of a one-dimensional Brauer–Severi variety is the projective conic section $Q$:

$$x _ {0} ^ {2} + x _ {1} ^ {2} + x _ {2} ^ {2} = 0$$

on the real projective plane $\mathbf P _ {\mathbf R } ^ {2}$. Over the field of complex numbers $\mathbf C$ this variety is isomorphic to the projective line $\mathbf P _ {\mathbf C } ^ {1}$. The set of all one-dimensional Brauer–Severi varieties, considered up to isomorphism, is in a one-to-one correspondence with the set of projective non-degenerate conic sections (considered up to projective equivalence over $k$), which is in turn in a one-to-one correspondence with the set of non-isomorphic generalized quaternion algebras over $k$. In the above example the conical section $Q$ corresponds to the algebra of ordinary quaternions.

In the more-dimensional case, the set of classes of $n$- dimensional Brauer–Severi varieties (i.e. Brauer–Severi varieties distinguished up to $k$- isomorphism) may be identified with the Galois cohomology group $H ^ {1} (k, \mathop{\rm PGL} (n + 1, k))$ where $\mathop{\rm PGL} (n + 1, k)$ is the projective group of automorphisms of the projective space $\textrm{ P } _ {k} ^ {n}$[3], [4]. This cohomology group describes the classes of $k$- isomorphic central simple $k$- algebras of rank $(n + 1) ^ {2}$( i.e. forms of the matrix algebra $M _ {n+1} (k)$). The connection between Brauer–Severi varieties and central simple algebras is more explicitly described as follows. To a $k$- algebra $A$ of rank $r ^ {2}$ one associates the variety $X$ of its left ideals of rank $r$, which is defined as a closed subvariety of the Grassmann manifold of all $k$- linear subspaces of dimension $r$ in $A$. In certain cases the variety $X$ may be defined by norm equations — e.g. in the case of quaternion algebras. The connection between Brauer–Severi varieties and algebras is taken advantage of in the study of both [1], [4].

The most significant properties of Brauer–Severi varieties are the following. A Brauer–Severi variety is $k$- isomorphic to a projective space $\mathbf P _ {k} ^ {n}$ if and only if it has a point in the field $k$. All Brauer–Severi varieties have a point in some finite separable extension $K$ of $k$[1].

The Hasse principle applies to Brauer–Severi varieties defined over an algebraic number field.

The field of rational functions $k(X)$ on a Brauer–Severi variety $X$ is the splitting field of the corresponding algebra $A$; moreover, an arbitrary extension $K$ of $k$ is the splitting field for $A$ if and only if $X$ has a $K$- point [4].

In the context of the generalization of the concepts of a central simple algebra and the Brauer group to include schemes, the Brauer–Severi varieties were generalized to the concept of Brauer–Severi schemes [2]. Let $f: P \rightarrow X$ be a morphism of schemes. A scheme $P$ is called a Brauer–Severi scheme if it is locally isomorphic to a projective space ${\mathbf P } _ {X} ^ {n}$ over $X$ in the étale topology of $X$. A scheme $P$ over a scheme $X$ is a Brauer–Severi scheme if and only if $f: P \rightarrow X$ is a finitely-presented proper flat morphism and if all of its geometrical fibres are isomorphic to projective spaces [2].

#### References

 [1] F. Châtelet, "Variations sur un thème de H. Poincaré" Ann. Sci. École Norm. Sup. (3) , 61 (1944) pp. 249–300 [2] A. Grothendieck, "Le groupe de Brauer" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 1–21 [3] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) [4] P. Roguette, "On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras" Math. Ann. , 150 (1963) pp. 411–439

Thus a Brauer–Severi variety of dimension $n$ is a $\overline{k}\; /k$- form of $\mathbf P _ {k} ^ {n}$.

How to Cite This Entry:
Brauer-Severi variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer-Severi_variety&oldid=46157
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article