hyper-resolutions

[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 $X_A\to \mathbf{max}R$ This situation is
very similar to that of a desingularization $X\to \mathbf{max}R$ 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
$M^{\theta }(Q,\alpha )\to {\mathbf{iss}}_{\alpha }Q$ 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 $X_{A(Q,\alpha )}\to {\mathbf{iss}}_{\alpha }Q$ 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 $A=\left[\begin{array}{cc}C[x,y]& C[x,y]\\ (x,y)& C[x,y]\end{array}\right]$ which is the quiver order of the quiver setting
$(Q,\alpha )$ then the Brauer-Severi fibration
$X_A\to {\mathbf{iss}}_{\alpha }Q$ 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 $B=\left[\begin{array}{cc}C[x,y]& C[x,y]\\ C[x,y]& C[x,y]\end{array}\right]$
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]: https://lievenlb.local/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/