Last time we saw that for $A$ a smooth order with center $R$ the Brauer-Severi variety $XA$ is a smooth variety and we have a projective morphism $XA \rightarrow \mathbf{max}~R$ 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) 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) of the Popov-conjecture for quiver-quotients (see for example his Ph.D. thesis Nullcones of quiver representations).
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 : 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 $~(Q,\alpha)$ then the Brauer-Severi fibration $~XA \rightarrow \mathbf{iss}{\alpha}~Q$ 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 $~XB \simeq \mathbb{A}^2 \times \mathbb{P}^1$. It turns out that $~XA$ is a blow-up of $~X_B$ at a point in the fiber over the zero-representation.

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Posted in geometry
Written on Thu, 30 September 2004 at 4:07 pm
Tags: Azumaya, Brauer, Brauer-Severi, moduli, non-commutative, representations
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