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Classical Brauer-Severi varieties can be described either as twisted

forms of projective space (Severi\’s way) or as varieties containing

splitting information about central simple algebras (Brauer\’s way). If

$K$ is a field with separable closure $\overline{K}$, the first approach

asks for projective varieties $X$ defined over $K$ such that over the

separable closure $X(\overline{K}) \simeq

\mathbb{P}^{n-1}_{\overline{K}}$ they are just projective space. In

the second approach let $\Sigma$ be a central simple $K$-algebra and

define a variety $X_{\Sigma}$ whose points over a field extension $L$

are precisely the left ideals of $\Sigma \otimes_K L$ of dimension $n$.

This variety is defined over $K$ and is a closed subvariety of the

Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = M_n(K)$ one

can use the matrix-idempotents to show that the left ideals of dimension

$n$ correspond to the points of $\mathbb{P}^{n-1}_K$. As for any central

simple $K$-algebra $\Sigma$ we have that $\Sigma \otimes_K \overline{K}

\simeq M_n(\overline{K})$ it follows that the varieties $X_{\Sigma}$ are

among those of the first approach. In fact, there is a natural bijection

between those of the first approach (twisted forms) and of the second as

both are classified by the Galois cohomology pointed set

$H^1(Gal(\overline{K}/K),PGL_n(\overline{K}))$ because

$PGL_n(\overline{K})$ is the automorphism group of

$\mathbb{P}^{n-1}_{\overline{K}}$ as well as of $M_n(\overline{K})$. The

ringtheoretic relevance of the Brauer-Severi variety $X_{\Sigma}$ is

that for any field extension $L$ it has $L$-rational points if and only

if $L$ is a _splitting field_ for $\Sigma$, that is, $\Sigma \otimes_K L

\simeq M_n(\Sigma)$. To give one concrete example, If $\Sigma$ is the

quaternion-algebra $(a,b)_K$, then the Brauer-Severi variety is a conic

$X_{\Sigma} = \mathbb{V}(x_0^2-ax_1^2-bx_2^2) \subset \mathbb{P}^2_K$

Whenever one has something working for central simple algebras, one can

_sheafify_ the construction to Azumaya algebras. For if $A$ is an

Azumaya algebra with center $R$ then for every maximal ideal

$\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central

simple $R/\mathfrak{m}$-algebra. This was noted by the

sheafification-guru [Alexander Grothendieck][2] and he extended the

notion to Brauer-Severi schemes of Azumaya algebras which are projective

bundles $X_A \rightarrow \mathbf{max}~R$ all of which fibers are

projective spaces (in case $R$ is an affine algebra over an

algebraically closed field). But the real fun started when [Mike

Artin][3] and [David Mumford][4] extended the construction to suitably

_ramified_ algebras. In good cases one has that the Brauer-Severi

fibration is flat with fibers over ramified points certain degenerations

of projective space. For example in the case considered by Artin and

Mumford of suitably ramified orders in quaternion algebras, the smooth

conics over Azumaya points degenerate to a pair of lines over ramified

points. A major application of their construction were examples of

unirational non-rational varieties. To date still one of the nicest

applications of non-commutative algebra to more mainstream mathematics.

The final step in generalizing Brauer-Severi fibrations to arbitrary

orders was achieved by [Michel Van den Bergh][5] in 1986. Let $R$ be an

affine algebra over an algebraically closed field (say of characteristic

zero) $k$ and let $A$ be an $R$-order is a central simple algebra

$\Sigma$ of dimension $n^2$. Let $\mathbf{trep}_n~A$ be teh affine variety

of _trace preserving_ $n$-dimensional representations, then there is a

natural action of $GL_n$ on this variety by basechange (conjugation).

Moreover, $GL_n$ acts by left multiplication on column vectors $k^n$.

One then considers the open subset in $\mathbf{trep}_n~A \times k^n$

consisting of _Brauer-Stable representations_, that is those pairs

$(\phi,v)$ such that $\phi(A).v = k^n$ on which $GL_n$ acts freely. The

corresponding orbit space is then called the Brauer-Severio scheme $X_A$

of $A$ and there is a fibration $X_A \rightarrow \mathbf{max}~R$ again

having as fibers projective spaces over Azumaya points but this time the

fibration is allowed to be far from flat in general. Two months ago I

outlined in Warwick an idea to apply this Brauer-Severi scheme to get a

hold on desingularizations of quiver quotient singularities. More on

this next time.

[1]: http://www.neverendingbooks.org/DATA/brauer.jpg

[2]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html

[3]: http://www.cirs-tm.org/researchers/researchers.php?id=235

[4]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html

[5]: http://alpha.luc.ac.be/Research/Algebra/Members/michel_id.html