[Last time][1] we saw that for $A$ a smooth order with center $R$ the

Brauer-Severi variety $X_A$ is a smooth variety and we have a projective

morphism $X_A \rightarrow \mathbf{max}~R$ This situation is

very similar to that of a desingularization $~X \rightarrow

\mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$.

The top variety $~X$ is a smooth variety and there is a Zariski open

subset of $~\mathbf{max}~R$ where the fibers of this map consist of just

one point, or in more bombastic language a $~\mathbb{P}^0$. The only

difference in the case of the Brauer-Severi fibration is that we have a

Zariski open subset of $~\mathbf{max}~R$ (the Azumaya locus of A) where

the fibers of the fibration are isomorphic to $~\mathbb{P}^{n-1}$. In

this way one might view the Brauer-Severi fibration of a smooth order as

a non-commutative or hyper-desingularization of the central variety.

This might provide a way to attack the old problem of construction

desingularizations of quiver-quotients. If $~Q$ is a quiver and $\alpha$

is an indivisible dimension vector (that is, the component dimensions

are coprime) then it is well known (a result due to [Alastair King][2])

that for a generic stability structure $\theta$ the moduli space

$~M^{\theta}(Q,\alpha)$ classifying $\theta$-semistable

$\alpha$-dimensional representations will be a smooth variety (as all

$\theta$-semistables are actually $\theta$-stable) and the fibration

$~M^{\theta}(Q,\alpha) \rightarrow \mathbf{iss}_{\alpha}~Q$ is a

desingularization of the quotient-variety $~\mathbf{iss}_{\alpha}~Q$

classifying isomorphism classes of $\alpha$-dimensional semi-simple

representations. However, if $\alpha$ is not indivisible nobody has

the faintest clue as to how to construct a natural desingularization of

$~\mathbf{iss}_{\alpha}~Q$. Still, we have a perfectly reasonable

hyper-desingularization $~X_{A(Q,\alpha)} \rightarrow

\mathbf{iss}_{\alpha}~Q$ where $~A(Q,\alpha)$ is the corresponding

quiver order, the generic fibers of which are all projective spaces in

case $\alpha$ is the dimension vector of a simple representation of

$~Q$. I conjecture (meaning : I hope) that this Brauer-Severi fibration

contains already a lot of information on a genuine desingularization of

$~\mathbf{iss}_{\alpha}~Q$. One obvious test for this seemingly

crazy conjecture is to study the flat locus of the Brauer-Severi

fibration. If it would contain info about desingularizations one would

expect that the fibration can never be flat in a central singularity! In

other words, we would like that the flat locus of the fibration is

contained in the smooth central locus. This is indeed the case and is a

more or less straightforward application of the proof (due to [Geert Van

de Weyer][3]) of the Popov-conjecture for quiver-quotients (see for

example his Ph.D. thesis [Nullcones of quiver representations][4]).

However, it is in general not true that the flat-locus and central

smooth locus coincide. Sometimes this is because the Brauer-Severi

scheme is a blow-up of the Brauer-Severi of a nicer order. The following

example was worked out together with [Colin Ingalls][5] : Consider the

order $~A = \begin{bmatrix} C[x,y] & C[x,y] \\ (x,y) & C[x,y]

\end{bmatrix}$ which is the quiver order of the quiver setting

$~(Q,\alpha)$ $\xymatrix{\vtx{1} \ar@/^2ex/[rr] \ar@/^1ex/[rr]

& & \vtx{1} \ar@/^2ex/[ll]} $ then the Brauer-Severi fibration

$~X_A \rightarrow \mathbf{iss}_{\alpha}~Q$ is flat everywhere except

over the zero representation where the fiber is $~\mathbb{P}^1 \times

\mathbb{P}^2$. On the other hand, for the order $~B =

\begin{bmatrix} C[x,y] & C[x,y] \\ C[x,y] & C[x,y] \end{bmatrix}$

the Brauer-Severi fibration is flat and $~X_B \simeq \mathbb{A}^2 \times

\mathbb{P}^1$. It turns out that $~X_A$ is a blow-up of $~X_B$ at a

point in the fiber over the zero-representation.

[1]: http://www.neverendingbooks.org/index.php?p=342

[2]: http://www.maths.bath.ac.uk/~masadk/

[3]: http://www.win.ua.ac.be/~gvdwey/

[4]: http://www.win.ua.ac.be/~gvdwey/papers/thesis.pdf

[5]: http://kappa.math.unb.ca/~colin/

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