Let $A$ be a quantum $\mathbb{P}^n$ defined by 
$$
A=\mathbb{C}\langle x\_1,x\_2,\dots,x\_{n+1}\rangle/(x\_ix\_j-r\_{ij}x\_jx\_i)\_{1\le i < j\le n+1}.
$$
I would like to know the set $X$ of isomorphism classes of point modules for $A$. Here a point module is a cyclic graded right $A$-module $M$ such that each graded piece of $M$ is one-dimensional. 


The set $Y$ of isomorphism classes of of point modules for the quantum $\mathbb{P}^2$
$$
\mathbb{C}\langle x\_1,x\_2,x\_3\rangle/(x\_ix\_j-r\_{ij}x\_jx\_i)\_{1\le i < j\le 3}
$$
is known to be $\mathbb{P}^2$ or a union of three lines in $\mathbb{P}^2$. 


It seems known that $X$ is projective for $n=4$, but does anyone know explicit description of $X$? I tried to compute $X$ in a similar manner as the quantum $\mathbb{P}^2$ case, but it seems quite complicated. I would also appreciate it if someone give me a good description for higher dimensional $n$ case (especially $n=3,4$). 


Thank you very much. 



Point modules of quantum projective space $\mathbb{P}^n$
$X$ is either isomorphic to $\mathbb{P}^n$ or is the union of some faces of the fundamental $n$-simplex (on the points $v\_i = [\delta\_{1i} : ... : \delta\_{n+1 i}]$) containing all $\mathbb{P}^1$'s making up the $1$-faces. The generic case corresponds to the collection of all these $\mathbb{P}^1$'s.


One proves this by induction, the essential reduction being that either $X$ is contained in the collection of hypersurfaces $x\_1.x\_2....x\_{n+1}=0$ or $A$ is the twisted homogeneous coordinate ring of (ordinary commutative) $\mathbb{P}^n$ (and hence $X \simeq \mathbb{P}^n$).


Indeed, let $x\_i$ act as a non-zero divisor on the point module $P$ then $P$ corresponds to a one-dimensional representation of the degree-zero part of the graded localization of $A$ at the normalizing element $x\_i$. This degree zero part is a quantum polynomial ring in $n$ variables. Either these variables all commute in which case $A$ is the claimed twisted coordinate ring or one of the new variables $y\_j = x\_jx\_i^{-1}$ vanishes on the one-dimensional representation and hence $x\_j$ vanishes on $P$.


If $X \subset \mathbb{V}(x\_1.x\_2... x\_{n+1})$ then one gets all point-modules on which $x\_i$ vanishes by looking at the point-modules of the quotient $A/(x\_i)$ which is again a quantum $\mathbb{P}^{n-1}$ and one gets the claimed $X$ by induction.


The induction starts with $n=1$ in which case $X=\mathbb{P}^1$ (as $A$ is a twisted coordinate ring) and $n=2$ in which case $X$ is either $\mathbb{P}^2$ or the triangle $x\_1x\_2x\_3=0$. So for $n=3$ one has $X$ either isomorphic to $\mathbb{P}^3$ or the union of the point-modules of the $4$ quantum $\mathbb{P}^2$'s determined by the quotients $A/(x\_i)$, each of these giving either a triangle or a $\mathbb{P}^2$. Etc.


