In dimension one, the SeveriâBrauer varieties are
conics. The corresponding central simple algebras are the
quaternion algebras. The algebra (a, b)K corresponds to the conic C(a, b) with equation
and the algebra (a, b)Ksplits, that is, (a, b)K is isomorphic to a
matrix algebra over K, if and only if C(a, b) has a point defined over K: this is in turn equivalent to C(a, b) being isomorphic to the
projective line over K.[1][2]
at the level of cohomology. Here H2(GL1) is identified with the
Brauer group of K, while the kernel is trivial because
H1(GLn) = {1}
by an extension of
Hilbert's Theorem 90.[3][4] Therefore, SeveriâBrauer varieties can be faithfully represented by Brauer group elements, i.e. classes of
central simple algebras.
Lichtenbaum showed that if X is a SeveriâBrauer variety over K then there is an exact sequence
Here the map δ sends 1 to the Brauer class corresponding to X.[2]
As a consequence, we see that if the class of X has order d in the Brauer group then there is a
divisor class of degree d on X. The associated
linear system defines the d-dimensional embedding of X over a splitting field L.[5]
Severi, Francesco (1932), "Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietĂ algebrica", Memorie della Reale Accademia d'Italia (in Italian), 3 (5), Reprinted in volume 3 of his collected works