k+1)\tau,q-(k +1)\tau)[-k -1] \longrightarrow  M(p,q) \longrightarrow  F(\omega+k \tau) \longrightarrow  0.
\end{align}
\qed
\eth

Note that when $k=0$, the four point modules $F(\omega)$ correspond to the four points $\{e_i\}_{0\le i\le 3}$ in the point scheme of $S$.  


Next, we recall some facts about the relationship between fat points and irreducible representations  of $S$. 
\bre{rem:simple} [$V(\omega+k \tau)$] Constructed originally by Sklyanin in \cite{Skly1983} for each $\omega\in E_2$ and $k\in \mathbb N\cup \{0\}$, there exists a representation $V(\omega+k\tau)$ \index{V0omega@$V(\omega+k\tau)$}
over $S$ in a certain $(k+1)$-dimensional subspace of theta functions of order $2k$. (We require $\Bbbk = \mathbb{C}$ here.) These representations were proved later by Smith and Staniszkis to be irreducible whenever $k<s$ \cite[Theorem 3.6]{SmithStan}. According to \cite[Proposition~3.3 and Section 6]{Smith1993}, all fat point modules $F(\omega+k\tau)$ of $S$ of multiplicity $<s$ arise as
\begin{align}\label{FatSimple}
F(\omega+k\tau)= \textstyle \bigoplus_{i\ge 0} V(\omega+k\tau)t^i,
\end{align}
where the graded $S$-action on the right side is given by $s_i\cdot v_jt^j=(s_iv_j)t^{i+j}$ for all $s_i\in S_i$ and $v_j\in V(\omega+k\tau)$. Here, the multiplicity of $F(\omega+k\tau)$ is equal to the dimension of $V(\omega+k\tau)$ which is equal to $k+1$. 
\ere

\bqu{rep-dims}
We can  extend \eqref{FatSimple} for $k=s-1$, where $F(\omega+(s-1)\tau)$ is a fat point module of multiplicity $s$. This yields 4 irreducible representations $V(\omega+(s-1)\tau)$ of dimension $s$ for $\omega\in E_2$. What are the forms of the other  irreducible representations of $S$ of dimension $s$?
\equ


Moreover, we have the following fact. Recall that $S$ is assumed to be module-finite over its center, so all of its irreducible representations are finite-dimensional. 

\ble{lem:quot} \cite[Lemma~4.1]{SmithStan}\cite[Sectiosn~3.3 and~5]{Smith1993}
Every irreducible representation $V$ of $S$ is the quotient of some fat point module $F$ of $S$. Here, $\dim_\Bbbk(V) \geq \text{mult}(F)$. Moreover, if $F$ is isomorphic to a shift $F[e]$ for some $e \geq 1$, then $\dim_\Bbbk(V)\leq e \cdot \text{mult}(F)$. \qed
\ele

\bnota{TwistM}[$V^\lambda$]
Let ${\rm Mod}_d(S)$ be the variety of all $d$-dimensional representations of $S$. The algebraic group ${\rm PGL}_d(\Bbbk)\times \Bbbk^\times$ acts on ${\rm Mod}_d(S)$ via $$((T,\lambda).\varphi_V)(a):=\lambda^iT\varphi_V(a)T^{-1}$$ for any $V\in {\rm Mod}_d(S)$ with corresponding map $\varphi_V: S\to {\rm End}(V)$ and $a\in S_i$. For any $V\in {\rm Mod}_d(S)$ and $\lambda\in \Bbbk^\times$, we define the {\it twisted representation} $$V^\lambda:=(1,\lambda).V.$$ It is important to point out that $V\cong V^\lambda$ if and only if there is some $T\in {\rm PGL}_d(\Bbbk)$ such that $(T,\lambda).V=V$.
\enota

 
\bpr{FatQS}  
Let $F$ be a fat point module over $S$ of multiplicity $d$, and $V$ be a nontrivial simple quotient of $F$. Then we have 
\begin{enumerate}
\item $\dim V=de$, where $e$ is the period of $F$ such that $F\cong F[e]$ and $F\not\cong F[i]$ for $1\le i\le e-1$ in $S$-qgr;
\item the stabilizer of $V$ in ${\rm PGL}_{de}(\Bbbk)\times \Bbbk^\times$ is conjugate to the subgroup generated by $(g_\zeta,\zeta)$ with $g_\zeta={\rm diag}(\underbrace{1,\dots,1}_{d},\underbrace{\zeta,\dots,\zeta}_{d},\dots,\underbrace{\zeta^{e-1},\dots,\zeta^{e-1}}_{d})$ and $\zeta$ is a primitive $e$-th root of unity;
\item all nontrivial simple quotients of $F$ are in the form of $V^{\lambda}$, for some irreducible representation $V$ of $S$ and for some $\lambda\in \Bbbk^\times$. \qed 
\end{enumerate}
\epr
\begin{proof}
Part (1) follows from \cite[Proposition 6 and its proof]{LeBruyn}.
Part (2) comes from \cite[Lemma~4]{BS}.
Part (3) holds by the proof of \cite[Theorem~3.13]{Smith1993}.
\end{proof}


We now turn our attention to the relationship between fat points and Azumaya locus of $S$, and for this we need some results from noncommutative projective algebraic geometry; see \cite{Artin:geometry} \cite[Section~5]{SmithTate} for further details. Let $$\mathcal R:={\rm Proj}(Z(S)).$$ We consider the sheaf $\mathcal S$ of $\mathcal O_\mathcal R$-algebras defined by $\mathcal S(\mathcal R_{(z)})=S[z^{-1}]_0$ for any non-zero homogeneous element $z\in Z(S)$.  Denote by $\mathcal Z$ the {\it central Proj} of $\mathcal S$, which satisfies  $\mathcal Z(\mathcal R_{(z)}) = Z(S[z^{-1}]_0)$ for any non-zero homogeneous element $z\in Z(S)$.  Consider the commutative scheme ${\bf Spec}(\mathcal Z)$ defined in \cite[Chapter~II, Exercise~5.17]{Hartshorne}, and we get by \cite[Theorem 5.2]{SmithTate} that it is isomorphic to ${\rm Proj}(Z(S^{(n/s)}))$. 
Now fat point modules of $S$ have support in ${\bf Spec}(\mathcal Z)$; indeed, a fat point module $F$ of $S$ gives rise to a sheaf $\mathcal F$ of $\mathcal S$-modules  such that $\mathcal F(\mathcal R_{(z)})=F[z^{-1}]_0$.

We get that there is a quotient morphism 
\begin{equation}\label{eq:SpecZ}
{\bf Spec}(\mathcal Z) \twoheadrightarrow \mathcal R
\end{equation} induced by the embedding $Z(S)\hookrightarrow Z(S^{(n/s)})$ via $z_i=u_i^{(n/s)}$ and $g_j=g_j$. 
We obtain that the support of a fat point module $F$ in $\mathcal R$ is the image of the support of the corresponding sheaf of $\mathcal S$-modules $\mathcal F$ in ${\bf Spec}(\mathcal Z)$ via \eqref{eq:SpecZ}. 

Now with \eqref{EOmega} and a result in \cite{SmithTate}, we obtain the following results on the fat point modules of $S$ of multiplicity $<s$.



\bpr{NAS}\cite[Theorem 5.11]{SmithTate} When $0 \leq k \leq s-1$, 
the support of the fat point modules $F(\omega+k\tau)$ and $F(\omega'+k'\tau)$ coincide in ${\bf Spec}(\mathcal Z)$ if $\omega+k\tau=-\omega'-k'\tau-2\tau$. \qed
\epr



\bnota{scriptC}[$\mathcal{C}(\omega+k\tau)$] Recall that for any $\omega\in E_2$ and $0\le k\le s-1$, the simple module $V(\omega+k\tau)$ constructed by Sklyanin  gives a non-zero point $(p_0,p_1,p_2,p_3,\gamma_1,\gamma_2)$ in $Y$ via its