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

Comments

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