text:
GST will cut inflation by 2%, boost India's economy: Hasmukh Adhia
Business PTI May 21, 2017 11:18:20 IST
Gulmarg: Inflation will fall by two percent on implementation of the goods and services tax (GST) and will create buoyancy in the economy, Revenue Secretary Hasmukh Adhia has said.
With the stage set for the biggest overhaul of India's tax system since Independence, the government will launch a massive awareness campaign to educate consumers about GST so that they are not fleeced by traders in the name of new tax.
In an interview to PTI, he said the all-powerful GST Council will meet next week to decide on tax rates of contentious items like gold, bidi and biscuits, just in time for its rollout from 1 July.
The Council over the two days last week assigned tax rates to more than 500 services and 1,200 goods by setting them in five broad rates of five, 12, 18 and 28 percent.
Adhia and his team visited this picturesque hill station near Srinagar for a short break before heading back to Delhi.
The rates, he said, have been so fixed that incidence of taxation has come down in many and remained at the same level as now in most of the remaining goods and services.
"I don't think inflation will at all go up because of GST. We have taken special care to ensure inflation does not go up. Our internal estimate is that after the rates are decided, inflation should come down by two percent," he said.
While the current indirect tax regime suffers from significant cascading, which leads to higher cost of goods and services, a free flow of credits across transactions under the GST framework will bring down the tax cost for businesses.
Also, taxpayers or consumers currently have to pay both the Centre and state taxes on a single sale, which adds to increased costs for businesses and consumers. Such an increase in costs adds to the inflationary pressure.
GST will be a single nation-wide sales tax replacing a string of central and state levies.
File image of Revenue Secretary Hasmukh Adhia. PTI
"That is how we have managed to keep our inflation basket under control," Adhia said, promising to fix any compliance issues that crop up during implementation.
The revenue secretary said GST will create buoyancy in the economy through better compliance and ease of doing business.
"I wouldn't say anything is pending, but I would say the government has to reach out to the trade and industry and also the machinery of explaining the GST procedures in town hall meetings. We need to accelerate this," he said on the task ahead.
Most importantly, consumer education campaign has to be taken up, he added.
"Because we have taken care to ensure the average tax incidence on commodities does not go up... There may be some traders who will try to tell consumers that under the changed GST rates, they will have to pay more. We have to educate them," he said.
Consumers, he stressed, need not be charged more in all cases even though the headline rate may go up as input tax credit or setting off the tax paid on raw material is available.
"We need a lot of consumer education for that," he said.
To safeguard consumer interest, the GST law provides for an anti-profiteering mechanism that will ensure industries that have got relief by way of lower taxes actually pass on the benefit to consumers.
"We will try to set it (anti-profiteering machinery) up as early as possible or we will try to identify an agency which will do it. We are working on it," Adhia said.
Even if setting up the agency takes three months, the agency can do a post mortem to see if anybody has unduly profited from GST.
"Even if it is set up after three months, it doesn't matter because if anybody has done profiteering, the books (of accounts) will speak for themselves and so, we can do a post mortem of those cases and we can take care of that," he said.
But the effort for now is to hold consultations with the trade and industry. "We will urge the big corporates that in the interest of smooth implementation of GST, they should not increase any rates now," he said.
The secretary said rates for all items but for six like gold, bidi, biscuits, textiles, footwear and natural or cultured pears and semi-precious stones have been decided.
"We decided to keep the things on which a lot of people have a lot of things to say - like bidi, gold, precious metal - and we will take up those at a later meeting. The main achievement now is we have been able to inform people about the tax rates of many of commodities except six," he added.
Adhia made the point that rates of many items have reduced while some have been put on the exemption list. "Our estimate is revenue loss on account of this will be more than offset by the revenue gain which we gain because of better compliance," he added.
He expressed confidence that the requirement of compensation to make up for revenue loss of any state will be fulfilled from cess on demerit and luxury goods like car and tobacco.
Adhia claimed that the IT network for GST is on track and there are no issues of privacy being compromised.
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Knowles Centre is proud to partner with the River East Transcona School Division and its John G. Stewart School.
Young people from Knowles Centre's Healing Homes and Day Treatment Programs attend John G. Stewart school, and staff from both organizations work closely for the benefit of the clients, but the relationship includes so much more.
Daily communications between the school and Knowles Centre allow all to track students' progress, address issues, and celebrate achievements on the way to student success.
Students benefit from the outstanding cooperation between Knowles Centre and John G. Stewart School around the Biimautaziiwin Aboriginal Cultural Program. Activities such as daily smudging, drumming, sweat lodge ceremonies, and traditional activities and celebrations are led by Knowles Centre's cultural coordinator and embraced by the school.
There is also a lot of collaboration around recreation. Most recently, art, sewing, and a bike repair program were jointly organized by staff from the school and Knowles Centre.
This partnership is also unique in that the school has the distinction of being the only publicly funded school on private property in Manitoba. This arrangement, and the goodwill between the two organizations, make it possible to often share resources. Knowles Centre is pleased to lend its camp for the annual Day Treatment camp each June. John G. Stewart School lends its gym to Knowles Centre for some recreation activities, and for the annual meeting.
It is not surprising that many in the community think that Knowles Centre and John G. Stewart School are part of the same organization. We see this as a testament to the positive relationship. However, any inquiries for the school should be made directly.
Jean-Paul Rochon
E-mail: jrochon@retsd.mb.ca
Or visit the John G. Stewart School webpage.

redpajama_set_name: RedPajamaCommonCrawl


Set in rural West Virginia, Mesha Maren's vibrant debut, "Sugar Run," is a novel of dreams and dark consequences. In 1989, Jodi McCarty is seventeen years old when she's sentenced to life in prison for manslaughter. She's released eighteen years later and finds herself at a Greyhound bus stop, reeling from the shock of unexpected freedom. Not yet able to return to her lost home in the Appalachian mountains, she goes searching for someone she left behind, but on the way, she meets and falls in love with Miranda, a troubled young mother. Together, they try to make a fresh start, but is that even possible in a town that refuses to change?


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Nato, Japan reaffirm support for Ukraine
Home/Politics/Bhim Army fighting to deliver justice to India's untouchables
Bhim Army fighting to deliver justice to India's untouchables
Online News EditorJune 8, 2022
By Moncho Torres
Haridwar, India, Jun 8 (EFE).- The untouchables or Dalits, members of the lowest strata of the Hindu caste system, are frequently victims of abuses, rape, lynching or community unrest in India, and when the authorities are reluctant to act, Bhim Army fights to deliver justice to them.
The Bhim Army, named after the historic Dalit leader Bhimrao Ramji Ambedkar, declares itself a non-violent movement, but employs intimidatory tactics to pressurize the authorities to act by mobilizing hundreds of its members en masse to places where an alleged injustice is committed.
A few weeks ago, a man, beating a drum at a village in northern India, announced that the local chief had ordered any untouchable found "on his property, in the crematorium (reserved for higher castes) or at the tube well," to be fined 5,000 rupees ($64) and beaten with shoes 50 times.
The video of the announcement soon became viral, and showed the prevalence of untouchability in the country despite it being prohibited in its Constitution, drafted in 1950 by a committee under Ambedkar.
Untouchability is a belief or practice whereby Dalits are considered impure and unclean, and are barred from accessing wells or temples. They are relegated to the most degrading menial tasks such as manual collection of feces, among others.
Chandra Shekhar Azad, the chief of Bhim Army and one of its founders in 2015, was quick to react.
He denounced the video on social media and said he would soon visit the village. Soon after, the police announced that they had arrested those involved, responding directly to the activist on Twitter.
"These incidents earlier never saw the light of the day but now when Bhim Army's mob goes there, it becomes both state and national news, it creates pressure, the administration is questioned. (…) It is because of us being present on the spot, the incidents get highlighted and justice is served to victims," Azad, 34, explained.
The Bhim Army leader greeted us in the sacred Hindu city of Haridwar, in the northern state of Uttarakhand, on the banks of the Ganges River, where he was training members of his group, which, by its estimates, has "tens of millions" of members across the country.
Azad was confident that due to the clear impact of his organization's work against injustice, a time will come when "in every home" of a Dalit family or other disadvantaged group in India "there will be a member of the Bhim Army."
The interview with this icon of Dalit struggle lasted hours, as every response by the lawyer-turned-activist turned into a discourse on the problems of the untouchables and the caste system in India, and their future projects to change this millennia-old oppression.
"In a country, where the constitution gives equal rights to everyone, how can one be superior or inferior? (…) Till the time there is casteism in this country, there will be no brotherhood and we will keep fighting among ourselves (…) My concept is very clear. I want casteism to vanish," stressed Azad.
This Hindu caste system divides society on the basis of birth into four major groups by order of purity – Brahmins (priests), Kshatriyas (warriors), Vaisyas (traders) and Sudras (servants), which in turn are subdivided into hundreds of subcastes.
At the bottom of them all are the untouchables, who, according to the last census of 2011, make up around 166 million people in India, or 16.2 percent of the population.
The activist recalled that these Dalits are still killed for things such as touching the food of a superior caste person, or beaten for eating in front of him.
Above all the great personalities who fought against the caste system, such as Ambedkar, Jotiba Phule, Periyar, Kanshi Ram, and saint Ravidas – who imagined a kingdom called Begumpura where "none are third or second. All are one," Azad considers that the person who most influenced his life was his father.
His father, a teacher who named him after one of the great revolutionaries of the Indian independence movement, told "a lot of stories about the discrimination he suffered during his lifetime." He died of cancer in January 2013.
"I have learned the meaning of struggle from him. How one has to give up on pleasures, luxuries, even go hungry to get one's children educated," he said.
That is why for "Bhim Army the main task is to open schools in rural areas to educate the children of the poor" and thus promote the movement for "self-respect."
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text:
\section{Introduction}
\label{sec:1}
\subsection{Background}
We denote by $M_{\PP^2}(r,c_1,n)$ the moduli of semistable torsion free sheaves $E$ on $\PP^2$ with the Chern class $c(E)=(r,c_1,n)\in H^{\ast}(\PP^2,\Z)$.
In this paper we treat the case where $c_1=1$.
In this case semistability and stability for $E$ coincide.
When $n\ge r\ge2$, or $n\ge 2$ and $r=1$, the Picard number of $\MPrn$ is equal to $2$ and we have two birational morphisms from $\MPrn$, which is described below.

One is defined by J. Li \cite{L} for general cases.
We denote by $\MPrn_0$ the open subset of $\MPrn$ consisting of stable vector bundles.
The Uhlenbeck compactification $\Mvrn$ of $\MPrn_0$ is described set theoretically by $$\Mv(r,1,n)=\sqcup_{i\ge 0}M_{\PP^2}(r,1,n-i)_0\times S^i(\PP^2).$$
The map $\pi\colon\MPrn\to\Mvrn\colon E\mapsto \pi(E)$ is defined by
$$\pi(E):=(E^{\vee\vee},\text{Supp}(E^{\vee\vee}/E))\in M_{\PP^2}(r,1,n-i)_0\times S^i(\PP^2),$$
where $E^{\vee\vee}$ is the double dual of $E$ and $i$ is the length of $E^{\vee\vee}/E$.
In the case where $r=1$, this morphism is called the Hilbert-Chow morphism $\pi\colon (\PP^2)^{[n]}\to S^n(\PP^2)$ and it is a divisorial contraction when $n\ge 2$.
In the case where $r\ge 2$, this map is birational since it is an isomorphism on $\MPrn_0$ to its image.
It is shown that the codimension of the complement of $\MPrn_0$ is equal to 1 when $\MP(r,1,n-1)\neq\emptyset$ (cf. \cite[Proposition~3.23]{M1}).
Hence this map is a divisorial contraction.

The other one is defined by Yoshioka.
In his paper \cite{Y2} on moduli of torsion free sheaves on rational surfaces, he studied the following morphism $$\psi\colon \MPrn\to\MP(n+1,1,n).$$
For any $E\in\MP(r,1,n)$, $\psi(E)$ is defined by the exact sequence
\begin{equation}\label{ex1}
0\to \Ext_{\PP^2}^1(E,\mop)^{\vee}
\otimes\mop\to\psi(E)\to E\to 0,
\end{equation}
which is called the universal extension, where $\Ext_{\PP^2}^1(E,\mop)^{\vee}$ is the dual vector space of $\Ext_{\PP^2}^1(E,\mop)$.
Here we have $\Homp(E,\mop)=\Extp^2(E,\mop)=0$ and $(n+1,1,n)\in H^{\ast}(\PP^2,\Z)$ is the Chern class of $$[E]-\chi(E,\mop)[\mop]=[E]+\dim\Extp^1(E,\mop)[\mop]\in K(\PP^2),$$ where $\chi(E,\mop)=\sum_i(-1)^i\dim_\C\Ext^i_{\PP^2}(E,\mop)$.

Furthermore the moduli space $\MPrn$ has
a stratification
$$\MPrn=\sqcup_{i=0}^r M^i_{\PP^2}(r,1,n),$$
where
$M^i_{\PP^2}(r,1,n):=\{E\in\MPrn\mid\dim_{\C}
\Hom_{\PP^2}(\mo_{\PP^2},E)=i\}$
and it is called the \emph{Brill-Noether locus}.
The following theorem is shown in \cite{Y2}.
\begin{theo}\emph{\bf cf. {\cite[Theorem~5.8]{Y2}}}
\label{yos}
The following hold.
\begin{itemize}
\item[(1)] There exists an
isomorphism $$M_{\PP^2}^i(r,1,n)
\cong\psi^{-1}\left(M^{n-r+i+1}_{\PP^2}(n+1,1,n)\right).$$
\item[(2)] The restriction of $\psi$ to each strata
$M_{\PP^2}^i(r,1,n)$ is a $Gr(n-r+i+1,i)$-bundle
over the strata $M^{n-r+i+1}_{\PP^2}(n+1,1,n)$.
\end{itemize}
\end{theo}
By the above theorem if $n$ is large enough, $\psi$ is a birational
morphism to the image $\im\psi$ and it is a flipping contraction.
By the theory of the birational geometry
\cite{BCHM} we have the diagram called flip
\begin{equation}\label{dai1}
\xymatrix{
\ar[dr]_{\psi_+}M_+(r,1,n)&&\ar@{-->}[ll]\MP(r,1,n).\ar[dl]
^{\psi}\\
&\im\psi&
}
\end{equation}

The purpose of this note is to describe
spaces $M_+(r,1,n)$, $\im\psi$ and the
morphism $\psi_+$ in the above diagram
using terms of moduli spaces.
We follow ideas in \cite{O}. We consider $M_{\PP^2}(r,1,n)$
as a moduli scheme of semistable modules over
the finite dimensional algebra
$\End_{\PP^2}\left(\mo_{\PP^2}(1)\oplus
\Omega_{\PP^2}(3)\oplus\mo_{\PP^2}(2)\right)$
and study the wall-crossing
phenomena as the stability changes
using the result of \cite{O} as follows.

\subsection{Main results}
We introduce the exceptional collection
$$\mathfrak{E}:=\left(\mo_{\PP^2}(1),
\Omega_{\PP^2}^1(3), \mo_{\PP^2}(2)\right)$$
on $\PP^2$ and put $\E:=\mo_{\PP^2}(1)\oplus\Omega_{\PP^2}^1(3)
\oplus\mo_{\PP^2}(2)$
and $B:=\End_{\PP^2}
(\E).$ We denote abelian categories of coherent sheaves on $\PP^2$ and finitely generated right $B$-modules by $\Coh(\PP^2)$ and $\Mod B$ respectively.
Then by Bondal's Theorem \cite{Bo}, the functor $\Phi:=\mathbf R \Hom_{\PP^2}(\E,-)$ gives an equivalence
$$\Phi\colon
\D(\PP^2)\cong \D(B),$$
where $\D(\PP^2)$ and $\D(B)$ are the bonded derived categories of $\Coh(\PP^2)$ and $\Mod B$ respectively.
The equivalence $\Phi$ also induces an isomorphism $\varphi\colon K(\PP^2)\cong K(B)$
between the Grothendieck groups
of $\Coh(\PP^2)$ and $\Mod B$.

For $\alpha\in K(B)$, we put $$\alpha^{\perp}:=\{\theta\in\Hom_\Z(K(B),\R)\mid\theta(\alpha)=0\}.$$
Any $\theta\in\alphap$ defines a stability condition of $B$-modules $E$ with $[E]=\alpha$.
We denote by $M_B(\alpha,\theta)$ the moduli space of $\theta$-semistable $B$-modules $E$ with $[E]=\alpha$.
In particular we take $$\alpha_{r}=\alpha_{r,n}:=\varphi(n\mo_{\PP^2}(-1)[2]+(2n-r+1)\mo_{\PP^2}[1]+(n-1)\mo_{\PP^2})\in K(B).$$
Here we omit subscription "$n$" although $\alpha_r$ depends on $n$, since we almost always fix $n$ in this paper.
There exists a wall-and-chamber structure on $\alpha_r^{\perp}$.

When $n$ is large enough we find two chambers $C_-, C_+$ and a wall $W_0\subset\alpha_{r}^{\perp}$ between them such that the following propositions hold (cf. \S~\ref{sec:3}).
We put $$M_-(\alpha_r):=M_B(\alpha_r,\theta_-),
\ \ M_+(\alpha_r):=M_B(\alpha_r,\theta_+),
\ \ M_0(\alpha_r):=M_B(\alpha_r,\theta_0)$$
for any $\theta_-\in C_-$, $\theta_+\in C_+$
and $\theta_0\in C_0$.
\begin{prop}\emph{\bf{\cite[Main~Theorem~1.3~(iii)]{O}}}
\label{inpro1}
We have an isomorphism $$M_{\PP^2}(r,1,n)\cong
M_-(\alpha_r)\colon E\mapsto \Phi(E[1]).$$
\end{prop}
We automatically get the following diagram
\begin{equation}\label{dai2}
\xymatrix{M_+(\alpha_r)\ar[dr]^{f_+}
&&\ar[dl]_{f_-}M_-(\alpha_r).\\
&M_0(\alpha_r)&
}
\end{equation}
By analyzing this diagram we see that
diagrams (\ref{dai1}) and (\ref{dai2}) coincide up to
isomorphism. In particular we get the
following proposition.
\begin{prop}\label{inpro2}
We have isomorphisms
\begin{itemize}
\item[(1)] $M_0(\alpha_r)\cong\im\psi$ and\\
\item[(2)] $M_+(\alpha_r)\cong
M_+(r,1,n)$.
\end{itemize}
\end{prop}
Proofs of Proposition~\ref{inpro2} are given
in \S~\ref{subsec:3-1} for (1) and in \S~\ref{subsec:3-4}
for (2).
Using the $B$-module $S_0:=\Phi(\mo_{\PP^2}[1])$ we define
the Brill-Noether locus similar to one in Yoshioka's
theory,
$$M_-^i(\alpha_r)=\{E\in M_-(\alpha_r)\mid
\dim_{\C}\Hom_{B}(S_0,E)=i\},$$
$$M_+^i(\alpha_r)=\{E\in M_+(\alpha_r)\mid
\dim_{\C}\Hom_{B}(E,S_0)=i\}.$$
Our situation is similar to \cite{NY} and we have our main theorem.

\begin{theo}\label{inmain}
Assume $n\ge r+2$. Then for each $i$ the following hold.
\begin{itemize}
\item[(1)] The images $f_+(M^i_+(\alpha_r))$
and $f_-(M^i_-(\alpha_r))$ coincide
in $M_0(\alpha_r)$.\\

We put $M^i_0(\alpha_r):=f_+(M^i_+(\alpha_r))
=f_-(M^i_-(\alpha_r))$.\\

\item[(2)] We have isomorphisms
$M_0^i(\alpha_r)\cong M_-^0(\alpha_{r-i})
\cong M_+^0(\alpha_{r-i})$.
\item[(3)] We have isomorphisms
$M_+^i(\alpha_r)\cong f^{-1}_+(M_0^i(\alpha_r))$.\\
\item[(4)] The restrcition of $f_+$ to each stratum
$M_+^i(\alpha_r)\to M_0^i(\alpha_r)$
is a $Gr(n-r+i-2,i)$-bundle over $M_0^i(\alpha_r)$.
\end{itemize}
\end{theo}
Note that $M_+(\alpha_r)\neq\emptyset$ if and only if $n\ge r+2$.
Proofs of Main Theorem~\ref{inmain} are given in \S~\ref{subsec:3-3} for (1) and \S~\ref{subsec:3-6} for the others.
We also give a new proof of Theorem~\ref{yos} using terms of $B$-modules via the isomorphism $M_{\PP^2}(r,1,n)\cong M_-(\alpha_r)$ in \S~\ref{subsec:3-6}.
By these descriptions we see that $M_+(r,1, n)$ is smooth and we can compute Hodge polynomials of $M_+(r,1,n)$ from those of $M_{\PP^2}(r,1,n)$.

The paper is organized as follows.
In \S 2 we introduce a description of Picard group of $M_{\PP^2}(r,1,n)$ in terms of $\theta$-stability of right $B$-modules.
In \S 3 we study the wall-crossing phenomena of moduli of $\theta$-semistable right $B$-modules.
This is described as stratified Grassmann bundles and this gives a proof of Main Theorem~\ref{inmain}.
In the Appendix by using Bridgeland stability we give a proof of Proposition~\ref{p2}, which is similar to \cite[Main Theorem~5.1]{O}.\\

\noindent\emph{\bf{Notation.}}
We fix the following notation in the paper:\\
If $A$ is a matrix we denote by ${}^tA$ the transpose
of $A$. If $V$ is $\C$-vector space then we denote
by $V^{\vee}$ the dual vector space $\Hom_{\C}(V,\C)$
of $V$ and we also denote by $Gr(V,i)$ the Grassmann
manifold of $i$-dimensional subspaces of V.
We consider the polynomial ring $\C[x_0,x_1,x_2]$ and the tensor product $V\otimes \C[x_0,x_1,x_2]$with a vector space $V$.
For any monomial $m\in\C[x_0,x_1,x_2]$ we put $$V\otimes m:=\{v\otimes m\in V\otimes \C[x_0,x_1,x_2]\mid v\in V\}.$$
We put $\mathbf x:=(x_{0},x_1, x_2)$ and denote by $V\otimes\mathbf x$ the direct sum $$(V\otimes x_{0})\oplus(V\otimes x_{1})\oplus(V\otimes x_{2})$$ of $V$.
We denote the $i$-th embedding $V\to V\otimes\mathbf x$ and the $i$-th projection $V\otimes\mathbf x\to V$ by $x_i$ and $x_i^{\ast}$ for $i=0,1,2$, respectively.
For morphisms $f_{ij}\colon U\to V$ between vector spaces $U$ and $V$ for $i,j=0,1,2$, we denote by the matrix
$\begin{pmatrix}
f_{00}&f_{01}&f_{02}\\
f_{10}&f_{11}&f_{12}\\
f_{20}&f_{21}&f_{22}
\end{pmatrix}
$
the morphism $$f:=\sum_{i,j}x_i\circ f_{ij}\circ x^{\ast}_j\colon U\otimes\mathbf x\to V\otimes\mathbf x.$$
We also use similar notation for vector bundles.

For any path algebra of quiver with relations, we identify modules over the algebra and representations of the corresponding quiver with relations.

\section{Picard group of $M_{\PP^2}(r,1,n)$}
\label{sec:2}
We introduce an explicit description of the Picard group of $M_{\PP^2}(r,1,n)$ in terms of $B$-modules.

\subsection{Finite dimensional algebra $B$}\label{ex}
Finite dimensional algebra
$B=\End_{\PP^2}(\E)$ is written as a path algebra of the following quiver
with relations
$(Q,J)$, where $Q$ is defined as
$$Q:=\xymatrix{\stackrel{v_{-1}}{\bullet}&\ar[l]_{\gamma_i}\stackrel{v_0}{\bullet}&\ar[l]_{\delta_j}\stackrel{v_1}{\bullet},} (i,j=0,1,2)$$
and $J$ is generated by the following relations
\begin{equation}\label{rel}
\gamma_i\delta_j+\gamma_j\delta_i=0, (i,j=0,1,2).
\end{equation}
We identify categories $\D(\PP^2)$ and $\D(B)$ and groups $K(\PP^2)$ and $K(B)$ via $\Phi$ and $\varphi$.
For example, we denote $\mo_{\PP^2}(i-1)[2-i]$ and the corresponding simple $B$-module $$\C v_i=\Phi(\mo_{\PP^2}(i-1)[2-i])$$ by the same symbols $S_i$ for $i=-1,0,1$.

We put $e_i:=[S_i]\in K(B)$ ($i=-1,0,1$).
Then we have $$K(B)=\Z e_{-1}\oplus\Z e_0 \oplus\Z e_1.$$
We denote the dual base by $\{e^{\ast}_{-1},e^{\ast}_0, e^{\ast}_1\}$.
For $\alpha_{-1}, \alpha_0,\alpha_1\in\Z$, by
\begin{equation}
\label{alpha}
\alpha=\begin{pmatrix}
\alpha_{-1}\\
\alpha_0\\
\alpha_1
\end{pmatrix}\in K(B)
\end{equation}
we denote $\alpha=\alpha_{-1}e_{-1}+\alpha_0e_0+\alpha_1e_1\in K(B)$ and for $\theta^{-1}, \theta^0, \theta^{1}\in\R$, by $$\theta=(\theta^{-1},\theta^0,\theta^1)\in\Hom_{\Z}(K(B),\R),$$ we denote $\theta=\theta^{-1}e_{-1}^{\ast}+\theta^{0}e_{0}^{\ast}+\theta^{1}e_{1}^{\ast}\in\Hom_{\Z}(K(B),\R)$.

\subsection{Moduli of semistable $B$-modules}
For any $\alpha\in K(B)$ and $\theta\in\alpha^{\perp}\otimes\R\subset\Hom_{\Z}(K(B),\R)$, we define $\theta$-stability as follows. Here $$\alpha^{\perp}=\{\theta\in \Hom_{\Z}(K(B),\Z)\mid\theta(\alpha)=0\}.$$

\begin{defi}
A right $B$-module $E$ with $[E]=\alpha$ in $K(B)$ is said to be $\theta$-semistable if for any proper submodule $F\subset E$, the inequality $\theta(F)\ge\theta(E)=0$ holds. If the inequality is always strict, then $E$ is said to be $\theta$-stable.
\end{defi}

By $M_B(\alpha,\theta)$ we denote a moduli
scheme of $\theta$-semistable $B$-module
$E$ with $[E]=\alpha$.
We define wall and chamber structure on $\alpha^{\perp}\otimes\R$
as follows. Wall is a ray $W=\R_{\ge 0}\theta^W$ in
$\alpha^{\perp}\otimes\R$
satisfying that there exists a $\theta^W$-semistable $B$-module
$E$ such that $E$ has a proper submodule $F$ with
$[F]\notin\Q_{>0}\alpha$ in $K(B)$ and $\theta^W(F)=0$.
A chamber is a connected component
of $(\alpha^{\perp}\otimes\R)\setminus\cup W$, where $W$ runs over the
set of all walls in $\alpha^{\perp}\otimes\R$.
For any chamber $C\subset\alpha^{\perp}\otimes\R$, the moduli space
$M_B(\alpha,\theta)$ does not depend on the choice of $\theta\in C$.

Here we assume that
$\alpha\in K(B)$ is indivisible and
$\theta$ is not on any wall in
$\alpha^{\perp}$, then there exists a universal family $\U$
of $B$-modules on $M_B(\alpha,\theta)$
\begin{equation}\label{univ1}
\U:=\left(\U_{-1}\xrightarrow{\hspace{0.8cm}\gamma^{\ast}_i\hspace{0.8cm}}\U_0\xrightarrow{\hspace{0.8cm}\delta^{\ast}_j\hspace{0.8cm}}\U_1\right), (i,j=0,1,2)
\end{equation}
where $\U_{-1}$, $\U_0$ and $\U_1$ are vector bundles corresponding to vertices $v_{-1},v_0, v_1$ and $\gamma_i^{\ast}\colon \U_{-1}\to \U_0$, $\delta_j^{\ast}\colon \U_0\to \U_1$ are morphisms corresponding to arrows $\gamma_i$, $\delta_j$.

\subsection{Deformations of $B$-modules}
We take $\alpha\in K(B)$ defined by (\ref{alpha}).
For any $B$-module $E$ with $[E]=\alpha$, by choosing basis of $Ev_{-1}$, $Ev_0$ and $Ev_1$ we have an isomorphism
\begin{equation}\label{mode}
E\cong (\C^{\alpha_{-1}}\stackrel{C_i}{\to}\C^{\alpha_{0}}
\stackrel{D_j}{\to}\C^{\alpha_{1}}),
\end{equation}
where $C_i\in\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_{0}})$
and $D_j\in\Hom_{\C}(\C^{\alpha_{0}},\C^{\alpha_{1}})$
correspond to the action of $\gamma_i$ and $\delta_j$
respectively for $i,j=0,1,2$.
The pull back of the heart $\Mod B$ of the standard t-structure of $\D(B)$ by $\Phi$ is a full subcategory $\mathcal{A}:=\langle\mo(-1)[2], \mo[1],\mo(1)\rangle$ of $\D(\PP^2)$.
The following complex of
coherent sheaves on $\PP^2$
$$\mo(-1)^{\alpha_{-1}}\xrightarrow{\sum_iC_ix_i}
\mo^{\alpha_{0}}\xrightarrow{\sum_iD_jx_j}
\mo(1)^{\alpha_1}$$
corresponds to $E$ in (\ref{mode}) via the equivalence $\Phi$,
where $x_0, x_1, x_2$ are homogeneous coordinates
of $\PP^2$.
By \cite[Lemma~4.6~(1)]{O}, $\Ext^2_B(E,E)$ is isomorphic to
the cokernel of the map
\begin{multline}\label{ext}
\left(\bigoplus_i
\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_{0}})x_i\right)
\bigoplus\left(\bigoplus_j
\Hom_{\C}(\C^{\alpha_{0}},\C^{\alpha_{1}})x_j\right)\\
\stackrel{d}{\to}\bigoplus_{i\le j}
\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_{1}})x_ix_j,
\end{multline}
where the map $d$ is defined by
$$d(\sum_i\xi_i x_i, \sum_j\eta_j x_j)=\sum_{i,j}(D_j\xi_i+\eta_iC_j)x_ix_j$$ for $\xi_i\in\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_{0}})\text{ and }\eta_j\in\Hom_{\C}(\C^{\alpha_{0}},\C^{\alpha_{1}}), (i,j=0,1,2).$
We study the deformation functor $\mathcal{D}_E\colon$(Artin/$k$) $\to$ (Sets).
For any Artin local $k$-ring $R$, the set $\mathcal{D}_E(R)$ consists of right $R\otimes B$-modules $E^R$ $$E^R=(R^{\alpha_{-1}}\stackrel{C^R_i}{\to}R^{\alpha_0}\stackrel{D^R_j}{\to}R^{\alpha_1}), D^R_jC^R_i+D^R_iC^R_j=0$$ such that $C^R_i\equiv C_i, D^R_j\equiv D_j$ modulo $m_R$ for each $i,j=0,1,2$, where $C^R_i$ and $D^R_j$ is $R$-linear maps and $m_R$ is the maximal ideal of $R$. We show the following lemma.
\begin{lemm}\label{def}
The deformation functor $\mathcal{D}_E$ has an obstruction theory with values in $\Ext^2_B(E,E)$.
\end{lemm}
\begin{proof}
For any small extension $$0\to\mathfrak{a}\to R'\to R\to0$$ with $m_R\mathfrak{a}=0$ and $E^R=(C^R_i,D^R_j)\in\mathcal{D}_E(R)$, we write $C^R_i=C_i+\xi_i$ and $D^R_j=D_j+\eta_j$ for $\xi_i\in\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_0})\otimes m_R$ and $\eta_j\in\Hom_{\C}(\C^{\alpha_{0}}, \C^{\alpha_1})\otimes m_R$.
By the isomorphism $m_R\cong m_{R'}/\mathfrak{a}$, we have lifts $\xi'_i\in\Hom_{\C}(\C^{\alpha_{-1}}, \C^{\alpha_0})\otimes m_{R'}$ and $\eta'_j\in\Hom_{\C}(\C^{\alpha_{0}},\C^{\alpha_1})\otimes m_{R'}$ of $\xi_i$ and $\eta_j$ respectively.
We put $C^{R'}_j:=C_j+\xi'_i$, $D^{R'}_j:=D_j+\eta'_j$.

Since $D^{R'}_jC^{R'}_i+D^{R'}_iC^{R'}_j
\equiv D^R_jC^R_i+D^R_iC^R_j=0$ modulo
$\mathfrak{a}$, we have an element
$$\sum_{i\le j}\left(D^{R'}_jC^{R'}_i+D^{R'}_iC^{R'}_j
\right)x_ix_j\in\Hom_{\C}(\C^{\alpha_{-1}},\C^{\alpha_1})
x_ix_j\otimes \mathfrak{a}.$$
By the image of this element to the cokernel
of (\ref{ext}) tensored by $\mathfrak{a}$,
we define an element
$\mathfrak{o}(E^R)$ in
$\Ext_B^2(E,E)\otimes\mathfrak{a}$.
This defines a well-defined map $\mathfrak{o}\colon
\mathcal{D}_E(R)\to \Ext_B^2(E,E)\otimes\mathfrak{a}$
and we easily see that $E^R$ lifts to $\mathcal{D}_E(R')$ if and only if
$\mathfrak{o}(E^R)=0$.
\end{proof}

\subsection{Chambers $C_{\PP^2}$
and Picard group of $M_{\PP^2}(r,1,n)$}
We take \begin{equation}\label{ind}
\alpha_r=
\begin{pmatrix}
n\\2n+r-1\\n-1
\end{pmatrix}
\in K(B)
\end{equation}
such that $\ch(\alpha_r)=
-(r,1,\frac{1}{2}-n)$ and assume that $M_{\PP^2}(r,1,n)$
is not empty set. Then
there exists a chamber $C_{\PP^2}\subset
\alpha_r^{\perp}\otimes\R$ such that $\Phi(\ \cdot\ [1])$ induces an isomorphism
\begin{equation}
\label{isom2}
M_{\PP^2}(r,1,n)\cong M_B(\alpha_r,\theta)
\end{equation}
for any $\theta\in C_{\PP^2}$
(see \cite[Main~Theorem~5.1]{O} or \cite{P}).
The chamber $C_{\PP^2}$ is characterized as follows.
We put $\theta_{\PP^2}:=(-r-1,1,-1+r)$.
Then we have $\theta_{\PP^2}(\mo_x)=0$
for any skyscraper sheaf at $x\in\PP^2$.
The closure of $C_{\PP^2}$ contains
the ray $\R_{\ge 0}\theta_{\PP^2}$ and
for certain $\theta\in C_{\PP^2}$ we have
$\theta(\mo_x)>0$ and $M_B(\alpha_r,\theta)\neq\emptyset$.

If we put $\theta_0:=(-n+1,0,n)$,
then by \cite[Lemma~6.2]{O}
we have
\begin{equation}\label{chaeq}
\R_{>0}\theta_0+\R_{>0}\theta_{\PP^2}
\subset C_{\PP^2}.
\end{equation}
In the case where $r=1$, by \cite[Lemma~6.3~(2)]{O}
we have
$\R_{>0}\theta_0+\R_{>0}\theta_{\PP^2}=C_{\PP^2}$ for $n\ge 2$
In the case where $r\ge 2$, we will describe
the chamber in \S~\ref{subsec:3-4}.

Since $\alpha_r$ is indivisible
by the definition (\ref{ind}), for
any $\theta\in C_{\PP^2}$ we have
a universal family $\U$ on $M_B(\alpha_r, \theta)$ as
in (\ref{univ1}).
We define a homomorphism
from $\alpha_r^{\perp}$ to $\Pic\left(M_B(\alpha_r,\theta)
\right)$ by
\begin{equation*}
\rho(\mathbf m)=m_{-1}\det(\U_{-1})+
m_0\det(\U_0)+m_1\det(\U_1),
\end{equation*}
for $\mathbf m=(m_{-1}, m_{0}, m_{1})\in\alpha_r^{\perp}$.
By (\ref{isom2}) this
gives a homomorphism $\rho\colon\alpha_r^{\perp}\to
\Pic\left(M_{\PP^2}(r,1,n)\right)$.
Then by \cite{D} we have the following proposition.
\begin{prop}\label{pic}
The above map $\rho\colon\alpha_r^{\perp}\to\Pic
\left(M_{\PP^2}(r,1,n)\right)$ is an isomorphism.
Furthermore $\rho(-3\theta_{\PP^2})$ is
the canonical bundle of $M_{\PP^2}(r,1,n)$.
\end{prop}

\section{Proof of Main Theorem~\ref{inmain}}
\label{sec:3}
We put $\alpha_{r,n}:={}^t\left(n,2n-1+r,n-1\right)\in K(B)$.
In the following we omit "$n$" and put $\alpha_r=\alpha_{r,n}$ except in \S~\ref{subsec:3-4}.
Note that $\ch(\alpha_r)=-(r,1,\frac{1}{2}-n)$.
We put $\theta_0:=(-n+1,0,n)\in\alpha^{\perp}_r$ and consider $\theta_+:=\theta_0+\e(2n-1+r,-n,0)$ and $\theta_-:=\theta_0-\e(2n-1+r,-n,0)$ for $\e>0$ small enough such that $\theta_{\pm}$ lie on no wall.
We put $M_{\pm}(\alpha_r):=M_B(\alpha_r,\theta_{\pm})$ and $M_0(\alpha_r):=M_B(\alpha_r,\theta_0)$.
By (\ref{isom2}) and (\ref{chaeq}) we have an isomorphism
\begin{equation}\label{iso}
\MPrn\cong M_-(\alpha_r).
\end{equation}
By $C_\pm$ we denote the chamber containing $\theta_\pm$ respectively and put $W_0:=\R_{\ge 0}\theta_0$.
Since $\theta_-\in C_{\PP^2}$, we have $C_-=C_{\PP^2}$.
We automatically get the following diagram:
\begin{equation}
\label{thad}
\xymatrix{
\ar[dr]^{f_+}M_+(\alpha_r)&&M_-(\alpha_r).\ar[dl]_{f_-}\\
&M_0(\alpha_r)&
}
\end{equation}
In this section we see that this diagram is described by stratified Grassmann bundles and give a proof of Main Theorem~\ref{inmain}.

\subsection{Kronecker modules}
\label{subsec:3-1}
We consider the 3-Kronecker quiver, which has 2 vertices $v_{-1}, v_{1}$ and $3$ arrows $\beta_0,\beta_1,\beta_2$ from $v_{1}$ to $v_{-1}$ $$\begin{minipage}{6cm}\xymatrix{\stackrel{v_{-1}}{\bullet}&&\ar[ll]^{\beta_i}\stackrel{v_1}{\bullet}}\end{minipage}, (i=0,1,2).$$ and consider the path algebra $T$.
Any right $T$-module $G$ has a decomposition $G=Gv_{-1}\oplus Gv_1$ and actions of $\beta_i$ define linear maps $Gv_{-1}\to Gv_1$ for $i=0,1,2$. By abbreviation we define $\theta_0(G)\in\R$ by $$\theta_0(G):=(-n+1)\dim_\C Gv_{-1}+n\dim_\C Gv_1.$$
We denote by $K(T)$ the Grothendieck group of the abelian category of finitely generated right $T$-modules and take $\alpha_T:=n[\C v_{-1}]+(n-1)[\C v_1]\in K(T)$.
\begin{defi}
An right $T$-module $G$ with $[G]=\alpha_T\in K(T)$ is stable if and only if for any non-zero proper submodule $G'$ of $G$ we have an inequality $\theta_0(G')> 0$.
\end{defi}
We denote by $M_T(\alpha_T)$
the moduli space of stable $T$-modules
$G$ with $[G]=\alpha_T$.
For any $B$-module $E=\left(\C^n\stackrel{C_i}{\to}
\C^{2n-1+r}\stackrel{D_j}{\to}
\C^{n-1}\right)$, we define $T$-module $E_T$ by
$$E_T:=\left(\C^n\stackrel{A_i}{\to}\C^{n-1}\right),$$
where $C_i$ and $D_j$ are matrices with
suitable sizes and we define $A_i$ by
$A_i:=D_{i+2}C_{i+1}$ for each
$i\in\Z/3\Z$.
\begin{lemm}\label{lemkro}
For any $B$-module $E=\left(\C^n\stackrel{C_i}{\to}
\C^{2n-1+r}\stackrel{D_j}{\to}
\C^{n-1}\right)$,
the following hold.
\begin{itemize}
\item[(1)] $E$ is $\theta_0$-semistable if and only if
$E_T$ is stable.
\item[(2)] $E$ is $\theta_0$-stable if and only if
$E_T$ is stable and $$\Homr(E,S_0)
=\Homr(S_0,E)=0.$$
\item[(3)] The following are equivalent.
\begin{itemize}
\item[(a$-$)] $E$ is $\theta_-$-stable.
\item[(b$-$)] $E$ is $\theta_-$-semistable.
\item[(c$-$)] $E_T$ is stable and $\Homr(E,S_0)
=0$.
\end{itemize}
\item[(4)] The following are equivalent.
\begin{itemize}
\item[(a$+$)] $E$ is $\theta_+$-stable.
\item[(b$+$)] $E$ is $\theta_+$-semistable.
\item[(c$+$)] $E_T$ is stable and $\Homr(S_0,E)
=0$.
\end{itemize}
\end{itemize}
\end{lemm}
\begin{proof}
(1) For every submodule $F\subset E$, we have a submodule $F_T$ of $E_T$.
Conversely for any submodule $G'$ of $E_T$, we define a submodule $F$ of $E$ such that $F_T=G'$ as follows.
We put $Fv_{-1}:=G'v_{-1}$, $Fv_{1}:=G'v_{1}$ and $Fv_{0}:=\sum_i C_i(Fv_{-1})\subset Ev_0$.
By the relations (\ref{rel}) we have a submodule $F:=Fv_{-1}\oplus Fv_0\oplus Fv_1$ of $E$ and $\theta_0(F)=\theta_0(F_T)=\theta_0(G')$.
This yields the claim.\\
(2) For any non-zero proper submodule $F\subset E$, $\theta_0(F)=0$ if and only if $\dimv(F)=(0,l,0)$ or $(n,l,n-1)$ for $0<l<2n+r-1$.
There exists no such $F$ if and only if $\Hom_B(S_0,E)=\Hom_B(E,S_0)=0$.\\
(3) (a$-$)$\implies$(b$-$) It is trivial.
(b$-$)$\implies$(c$-$) We choose $\theta_-=\theta_0-\e(2n-1+r,-n,0)$ for $\e>0$ small enough.
If $E$ is $\theta_-$-semistable, then for any submodule $F\subset E$ we have $\theta_0(F)\ge 0$, since $\theta_-(F)\ge0$ for arbitrary small $\e>0$.
This implies that $E$ is $\theta_0$-semistable and hence by (1), $E_T$ is semistable.
Any non-zero $\phi\in\Hom_{B}(E,S_0)$ destabilize $E$.
Hence we also have $\Hom_{B}(E,S_0)=0$.\\
(c$-$)$\implies$(a$-$) We assume that $E_T$ is stable and $\Hom_{B}(E,S_0)=0$. Hence for every submodule $F\subset E$, we have $\theta_0(F)=\theta_0(F_T)\ge 0$.
If $\theta_0(F)= 0$ then $\Hom_{B}(E,S_0)=0$ implies that $F\cong S_0^{\oplus l}$ for $0\le l\le 2n+r-1$.
In this case $\theta_-(F)=\e nl>0$.
If $\theta_0(F)>0$ then we also have $\theta_-(F)>0$ for $\e$ small enough.
Hence $E$ is $\theta_-$-stable.\\
(4) It is similar to the proof of (3).\\
\end{proof}
By the above lemma we have morphisms $\pi_{\pm}^r\colon
M_{\pm}(\alpha_r)\to M_T(\alpha_T)\colon
E\mapsto E_T.$
We also see that the map $E\mapsto E_T$
is independent of representatives of S-equivalence
class for $\theta_0$-stability up to isomorphism
of $T$-modules.
Hence we get the morphism
$\pi_0^r\colon M_0(\alpha_r)\to M_T(\alpha_T)$
and this map is set theoretically injective.
\begin{lemm}\label{clo}
The morphism $\pi_0^r\colon M_0(\alpha_r)\to M_T(\alpha_T)$
gives a closed embedding.
\begin{proof}
$\pi_0^r$ is induced from a homomorphism of graded rings of
invariant sections, therefore affine morphism. Since both
of $M_0(\alpha_r)$ and $M_T(\alpha_T)$ are projective, $\pi_0^r$ is finite.
Since $\pi_0^r$ is set theoretically injective,
the claim holds.
\end{proof}
\end{lemm}

Furthermore we easily see that morphisms $$\pi_-^{n+1}\colon M_-(\alpha_{n+1})\to M_T(\alpha_T), \pi_+^{n-2}\colon M_+(\alpha_{n-2})\to M_T(\alpha_T)$$ are isomorphisms.
Inverse maps $$E_T=\left(\C^n\stackrel{A_i}{\to}\C^{n-1}\right)\mapsto E_\pm=\left(\C^n\stackrel{C_i^\pm}{\to}\C^{2n-1+r}\stackrel{D_j^\pm}{\to}\C^{n-1}\right)$$ of $\pi^{n-2}_+$ and $\pi^{n+1}_-$ are defined by
$$(C_0^+,C_1^+,C_2^+)=
\begin{pmatrix}
0&-A_2&A_1\\
A_2&0&-A_0\\
-A_1&A_0&0
\end{pmatrix}
,
\begin{pmatrix}
D_0^+\\D_1^+\\D_2^+
\end{pmatrix}
=I_{3n-3}
\text{ for }\pi^{n-2}_+$$
and
$$(C_0^-,C_1^-,C_2^-):=I_{3n},
\begin{pmatrix}
D^-_0\\D^-_1\\D^-_2
\end{pmatrix}
:=\begin{pmatrix}
0&-A_2&A_1\\
A_2&0&-A_0\\
-A_1&A_0&0
\end{pmatrix}
\text{ for }\pi^{n+1}_-,$$
where $I_{3n}$ and $I_{3n-3}$ are unit matrices with sizes $3n$ and $3n-3$ respectively.

Hence we get the diagram:
\begin{equation}\label{kro}
\xymatrix{
M_+(\alpha_r)\ar[dr]^{f_+}\ar[dd]_{g_+}&&
\ar[dd]^{g_-}\ar[dl]_{f_-}M_-(\alpha_r)\\
&M_0(\alpha_r)\ar@{->}[d]^{\pi_0^r}&\\
M_+(\alpha_{n-2})\ar[r]^{\cong}_{\pi_+^{n-2}}&
M_T(\alpha_T)&\ar[l]_{\cong}^{\pi_-^{n+1}}M_-(\alpha_{n+1})}
\end{equation}
where $g_-:=(\pi_-^{n+1})^{-1}\circ\pi_0^r\circ f_-$
and $g_+:=(\pi_+^{n-2})^{-1}\circ\pi_0^r\circ f_+$.
Morphisms $g_-$ and $g_+$ are explicitly defined by the following universal extensions for each $E_-\in M_-(\alpha_r)$ and $E_+\in M_+(\alpha_r)$,
\begin{equation}\label{ex2}
0\to \Ext^1_B(E_-,S_0)^{\vee}\otimes S_0\to g_-(E_-)\to E_-\to 0,
\end{equation}
\begin{equation}\label{ex3}
0\to E_+\to g_+(E_+)\to \Ext^1_B(S_0,E_+)\otimes S_0\to 0.
\end{equation}
Hence via isomorphisms (\ref{iso}), the morphism $g_-$ coincides with Yoshioka's map $\psi$, which is defined by a similar exact sequence (\ref{ex1}).
By Lemma~\ref{lemkro} and the diagram~(\ref{kro}), we have
$$M_0(\alpha_r)\cong \im g_-\cong \im \psi.$$
This gives a proof of (1) in Proposition~\ref{inpro2}.

\subsection{Brill-Noether locus}
\label{subsec:3-2}
We introduce
the Brill-Noether locus $M_-^i(\alpha_r)$ and
$M_+^i(\alpha_r)$ as follows.
$$M_-^i(\alpha_r):=\{E_-\in M_-(\alpha_r)\mid
\dim_{\C}\Hom_B(S_0,E_-)=i\},$$
$$M_+^i(\alpha_r):=\{E_+\in M_+(\alpha_r)\mid
\dim_{\C}\Hom_B(E_+,S_0)=i\}.$$
When we replace '=i' by '$\ge$ i'
in the right hand side, the corresponding
moduli spaces are denoted by the
left hand side with 'i' replaced by
'$\ge$ i'.

If we put
$\delta^{\ast}:=\sum_ix_i\circ\delta^{\ast}_i
\colon\U_0\to\U_1\otimes\mathbf x$,
then the zero locus of
$\wedge^{\rk\U_0-i+1}\delta^{\ast}$ defines
$M_-^{\ge i}(\alpha_r)$ as a closed subscheme
of $M_-(\alpha_r)$ because
$\ker \delta^{\ast}_x\cong \Hom_B(S_0,\U_x)$ for
any $x\in M_-(\alpha_r)$, where $\U=\left(
\U_{-1}\stackrel{\gamma_i^{\ast}}{\to}
\U_0\stackrel{\delta_j^{\ast}}{\to}
\U_1\right)$ is a
universal family of $B$-modules on $M_-(\alpha_r)$.
Similarly,
$M_+^{\ge i}(\alpha_r)$ is defined as a closed subscheme
of $M_+(\alpha_r)$. $M_-^i(\alpha_r)=M_-^{\ge i}(\alpha_r)
\setminus M_-^{\ge i+1}(\alpha_r)$ and
$M_+^i(\alpha_r)=M_+^{\ge i}(\alpha_r)
\setminus M_+^{\ge i+1}(\alpha_r)$ are open subset
of $M_-^{\ge i}(\alpha_r)$ and $M_+^{\ge i}(\alpha_r)$,
respectively.

\subsection{Set-theoretical description of
Grassmann bundles}
\label{subsec:3-3}
By Lemma~\ref{lemkro} we have the following
proposition.

\begin{prop}\label{set}
The following hold.
\begin{itemize}
\item[(1)] For any $E_-\in M_-^i(\alpha_r)$, we put $E':=\coker\left(\Hom_B(S_0,E_-)\otimes S_0\to E_-\right)$.
Then $E'$ is $\theta_-$-semistable and $\Hom_B(S_0,E')=0$, that is, $E'\in M_-^0(\alpha_{r-i})$. Hence $E'$ is also $\theta_0$-stable.
\item[(2)] Conversely, for any $E'\in M_-^0(\alpha_{r-i})$ and any $i$-dimensional vector subspace $V\subset\Ext^1_B(E',S_0)$, we obtain a $B$-module $E_-$ by the canonical exact sequence
$$0\to V^{\vee}\otimes S_0\to E_-\to E'\to 0.$$
Then $E_-$ is $\theta_-$-semistable and $\Hom_B(S_0,E_-)\cong V$, that is, $E_-\in M_-^i(\alpha_r)$.
\item[(3)] For any $E_+\in M_+^i(\alpha_r)$, we put $E':=\ker\left(E_+\to \Hom_B(E,S_0)^{\vee}\otimes S_0\right)$.
Then $E'$ is $\theta_+$-semistable and $\Hom_B(E',S_0)=0$, that is, $E'\in M_+^0(\alpha_{r-i})$. Hence $E'$ is also $\theta_0$-stable.
\item[(4)] Conversely, for any $E'\in M_+^0(\alpha_{r-i})$ and any $i$-dimensional vector subspace $V\subset\Ext^1_B(S_0,E')$, we obtain a $B$-module $E_+$ by the canonical exact sequence
$$0\to E'\to E_+\to V\otimes S_0\to 0.$$
Then $E_+$ is $\theta_+$-semistable and $\Hom_B(E_+,S_0)\cong V$, that is, $E_+\in M_+^i(\alpha_r)$.
\end{itemize}
\end{prop}
By Lemma~\ref{lemkro}, $M_-^0(\alpha_{r-i})$ is
set theoretically equal to $M_+^0(\alpha_{r-i})$.
For any $B$-module $E$, we have $\Ext^2_B(S_0,E)=
\Ext^2_B(E,S_0)=0$ by \cite[Lemma~4.6~(1)]{O}.
Hence by the Rieman-Roch formula,
for any element
$E'\in M_-^0(\alpha_{r-i})
=M_+^0(\alpha_{r-i})$ we have
$\dim_{\C}\Ext^1_B(E',S_0)=n+1-r+i$
and $\dim_{\C}\Ext^1_B(S_0,E')=
n-2-r+i$.
If $n-2-r\ge 0$, then by the above lemma
we have set theoretical equalities
\begin{equation}\label{seteq}
\begin{split}
f_-\left(M_-^i(\alpha_r)\right)
&=\left\{S_0^{\oplus i}\oplus E'\mid
E'\in M_-^0(\alpha_{r-i})=M_+^0(\alpha_{r-i})\right\}/\equiv_S\\
&=f_+\left(M_+^i(\alpha_r)\right),
\end{split}
\end{equation}
where $\equiv_S$ denotes the S-equivalence relation (cf. \cite[\S~4.1]{O}).
This gives a proof of (1) of Main~Theorem~\ref{inmain}.
Fibers of S-equivalence class of
$S_0^{\oplus i}\oplus E'$ by $f_-$ and $f_+$ are parametrized
by $Gr(\Ext^1_B(E',S_0),i)$ and $Gr(\Ext^1_B(S_0,E'),i)$
for $E'\in M_-^0(\alpha_{r-i})=M_+^0(\alpha_{r-i})$.

\begin{lemm}For any
integer $i>r$ the following holds.
$$M_-^i(\alpha_r)=M_+^i(\alpha_r)=\emptyset$$
\end{lemm}
\begin{proof}
By \cite[Lemma~5.7]{Y2}, we have
$M_-^i(\alpha_r)=\emptyset$ for any
$i>r$.
By (\ref{seteq}) this implies
$M_+^i(\alpha_r)=\emptyset$.
\end{proof}

\subsection{Description of $C_{\PP^2}$}
\label{subsec:3-4}
In the following proposition we use the symbol
$\alpha_{r,n}={}^t(n,2n-1+r,n-1)\in K(B)$.
\begin{prop}\label{cha}
The following hold.
\begin{itemize}
\item[(1)] $\MPrn\neq\emptyset$ if and only if
$n\ge r-1$.
\end{itemize}
In the following, we assume
$r\ge 2$.
\begin{itemize}
\item[(2)] $W_0=\R_{\ge 0}\theta_0$ is a wall on
$\alpha_{r,n}^{\perp}\otimes\R$ for $n\ge r-1$.
\item[(3)] $W_{\PP^2}=\R_{\ge 0}\theta_{\PP^2}$ is a wall on
$\alpha_{r,n}^{\perp}\otimes\R$ for $n\ge r$.
\end{itemize}
Hence we have $\R_{> 0}\theta_0+\R_{> 0}\theta_{\PP^2}
=C_{\PP^2}$ if $n\ge r$.
\end{prop}
\begin{proof}
(1) By the criterion for the existence of
non exceptional stable sheaves in \cite[\S~16.4]{P1},
we have our claim.\\
(2) We assume $n\ge r-1$.
By (1), there exists an element $E$ of
$M_-(\alpha_{r-1,n})\cong\MP(r-1,1,n)$.
By Lemma~\ref{lemkro}~(1),
a $B$-module $E\oplus S_0$
is $\theta_0$-semistable and has a submodule
$S_0$ with $\theta_0(S_0)=0$.
Hence $W_0=\R_{\ge 0}\theta_0$ is a wall on
$\alpha_{r,n}^{\perp}\otimes\R$.\\
(3) We assume $n\ge r$ and
take an element $\F$ of $\MP(r,1,n-1)$.
We consider the exact sequence
$$0\to\F'\to\F\to \mo_x\to0$$
for skyscraper sheaf $\mo_x$ at any
point $x\in\PP^2$.
Then $\F'$ is stable since $\mu(\F')=\mu(\F)$.
This gives elements $F:=\Phi(\F[1])$,
$F':=\Phi(\F'[1])$ of $M_-(\alpha_{r,n-1})$,
$M_-(\alpha_{r,n})$ respectively and an exact sequence
of $B$-modules
$$0\to\Phi(\mo_x)\to F'\to F\to0.$$
Hence $W_{\PP^2}=\R_{\ge0}\theta_{\PP^2}$ is
a wall on $\alpha_r^{\perp}\otimes\R$.
These together with (\ref{chaeq}) imply the last assertion.
\end{proof}
By this proposition and \cite[Lemma~6.3~(2)]{O} we have $\R_{> 0}\theta_0+\R_{> 0}\theta_{\PP^2}
=C_{\PP^2}$ if $r=1$ and $n\ge 2$, or $r\ge 2$ and $n\ge r$.
In this case $C_+$ is different from $C_-=C_{\PP^2}$ and it is adjacent to $C_-=C_{\PP^2}$ with the boundary containing $W_0$.
By the description of the canonical bundle of $M_-(\alpha_r)$ in Proposition~\ref{pic} we see that the diagram (\ref{thad}) gives the flip of $M_-(\alpha_r)$.
Hence we get an isomorphism $M_+(\alpha_r)\cong M_+(r,1,n)$ and a proof of (2) in Proposition~\ref{inpro2}.
\begin{prop}\label{sm}
Moduli schemes $M_-(\alpha_r)$ and $M_+(\alpha_r)$ are smooth.
\end{prop}
\begin{proof}
By Lemma~\ref{def}, deformation functors of $B$-modules $E$ have obstruction theories with values in $\Ext^2_{B}(E,E)$.
Since $\Ext^2_{B}(E_-,E_-)=0$ for any $E_-\in M_-(\alpha_r)\cong M_{\PP^2}(r,1,n)$ we see that $M_-(\alpha_r)$ is smooth (cf. \cite[Corollary~4.5.2]{HL}).
Furthermore if $E_+$ is an element of $M_+(\alpha_r)$ then by Proposition~\ref{set} we have an exact sequence
$$0\to E'\to E_+\to\C^i\otimes S_0\to 0$$ for some $i$ and $E'\in M_-^0(\alpha_{r-i})$.
Since $$\Ext^2_{B}(S_0,E_+)=\Ext^2_{B}(E',E')=\Ext^2_{B}(E',S_0)=0,$$ we also have $\Ext^2_B(E_+,E_+)=0$.
Thus $M_+(\alpha_r)$ is also smooth.
\end{proof}
In the rest of this section we show that the diagram~(\ref{thad}) is scheme theoretically described by stratified Grassmann bundles.

\subsection{Coherent systems}
\label{subsec:3-5}
For $r\ge i\ge 0$ we define moduli of coherent
systems $M_-(\alpha_r,i)$ and $M_+(\alpha_r,i)$ :
$$M_-(\alpha_r,i):=\{(E_-,V)\mid E_-\in
M_-(\alpha_r), V\subset\Hom_B(S_0,E_-)
\text{ with } \dim_{\C}V=i\},$$
$$M_+(\alpha_r,i):=\{(E_+,V)\mid E_+\in
M_+(\alpha_r), V\subset\Hom_B(E_+,S_0)
\text{ with } \dim_{\C}V=i\}.$$
These moduli schemes are constructed as follows.
We only show the construction of $M_-(\alpha_r,i)$
because the construction of $M_+(\alpha_r,i)$ is
similar.

We introduce the following quiver with relations
$(\bar{Q},I)$, where
$$\bar{Q}:=
\begin{minipage}{6cm}
$\xymatrix{
&&\stackrel{u}{\bullet}\ar[d]_{\rho}&&\\
\stackrel{\bar{v}_{-1}}{\bullet}&&\ar[ll]_{\bar{\gamma}_i}
\ar[d]_{\iota}\stackrel{\bar{v}_0}{\bullet}&&\ar[ll]_{\bar{\delta}_j}
\stackrel{\bar{v}_1}{\bullet},\\
&&\stackrel{w}{\bullet}&&}$
\end{minipage}
 (i,j=0,1,2)$$
and $I$ is generated by the following relations
$$\bar{\gamma}_i\rho=\iota\bar{\delta}_j=
\bar{\gamma}_i\delta_j+\bar{\gamma}_j\delta_i
=\iota\rho=0, (i,j=0,1,2).$$
Let $\bar{B}$ be a path algebra $\C\bar{Q}/I$
of the quiver with relations $(\bar{Q},I)$.
We have simple modules $\C \bar{v}_{-1}$,
$\C \bar{v}_{0}$,
$\C \bar{v}_{1}$, $\C u$ and $\C w$.
For each $\alpha_r\in K(B)$,
we put
$$\bar{\alpha}_r:=n[\C\bar{v}_{-1}]+(2n+r-1)[\C
\bar{v}_0]+(n-1)[\C\bar{v}_1]+(2n+r-i)[\C u]
+i[\C w]\in K(\bar{B}),$$
and for $\theta_-=(\theta_-^{-1},
\theta_-^{0},\theta_-^1)\in \alpha_r^{\perp}$
and $\e'>0$ small enough,
we put
$$\bar{\theta}_-:=\theta_-^{-1}[\C\bar{v}_{-1}]^{\ast}+
\theta_-^0[\C\bar{v}_0]^{\ast}+\theta_-^1[\C\bar{v}_1]^{\ast}
+\frac{\e'}{2n+r-i-1}[\C u]^{\ast}-\frac{\e'}{i}[\C w]^{\ast}
\in\bar{\alpha}_r^{\perp}.$$

For any right $\bar{B}$-module $\bar{E}_-$ with
$[\bar{E}_-]=\bar{\alpha}_r\in K(\bar{B})$,
$$\bar{E}_-=
\begin{minipage}{6cm}
$\xymatrix{
0\ar[rr]\ar@{}[drr]|\circlearrowleft&&
\C^{2n-1+r-i}\ar[rr]
\ar@{}[drr]|\circlearrowleft&&0\\
\ar@{}[drr]|\circlearrowleft
\ar[u]\C^n\ar[rr]^{\bar{C}_i}&&
\ar[u]^B\C^{2n-1+r}\ar[rr]^{\bar{D}_j}
\ar@{}[drr]|\circlearrowleft&&\ar[u]\C^{n-1}\\
\ar[u]0\ar[rr]&&
\ar[u]^{A}\C^i\ar[rr]
&&\ar[u]0}$
\end{minipage}
$$
we put
$$E_-:=
\begin{minipage}{6cm}
$\left(\xymatrix{
\C^n\ar[rr]^{\bar{C}_i}&&
\C^{2n-1+r}\ar[rr]^{\bar{D}_j}
&&\C^{n-1}}\right).$
\end{minipage}$$
The following lemma is proved similarly
as in Lemma~\ref{lemkro}~(3).
\begin{lemm}
If we take $\e'$ small enough, then
$\bar{E}_-$ is $\bar{\theta}_-$-semistable
if and only if $E_-$ is $\theta_-$-semistable
and $A$ is injective
and $B$ is surjective.
\end{lemm}
Hence if we denote by $M_{\bar{B}}(\bar{\alpha}_r,\bar{\theta}_-)$
the moduli of $\bar{\theta}_-$-semistable
$\bar{B}$-module $\bar{E}_-$ with $[\bar{E}_-]=
\bar{\alpha}_r$,
we get an isomorphism $M_{\bar{B}}(\bar{\alpha}_r,\bar{\theta}_-)
\cong M_-(\alpha_r,i)$. We write as $\bar{E}_-=(E_-,\C^i)
\in M_-(\alpha_r,i)$ by abbreviation.

We have morphisms
$$q_1\colon M_-(\alpha_r,i)\to M_-(\alpha_r)\colon
\bar{E}_-=(E_-,\C^i)\mapsto E_-$$ and
$$q_2\colon M_-(\alpha_r,i)\to
M_-(\alpha_{r-i})\colon \bar{E}_-\mapsto
q_2(\bar{E}_-)$$ defined by the canonical exact sequence
$$0\to\C^i\otimes S_0\to E_-\to q_2(\bar{E}_-)\to 0.$$

Similarly we have morphisms $q_1'\colon M_+(\alpha_r,i)
\to M_+(\alpha_r)$
and $q_2'\colon M_+(\alpha_r,i)\to M_+(\alpha_{r-i})$.
If we take an element $\bar{E}_+:=(E_+,\C^i)\in
M_+(\alpha_r,i)$, then
$q_1'$ and $q_2'$ are defined by $q_1'(\bar{E}_+)=E_+$
and $q_2'(\bar{E}_+):=\ker\left(E_+\to
(\C^i)^{\ast}\otimes S_0\right)$.
\begin{prop}\label{grbun}
The following hold.
\begin{itemize}
\item[(1)] The morphism $q_1\colon M_-(\alpha_r,i)\to M_-(\alpha_r)$ is a $Gr(j,i)$-bundle over each strata $M^j_-(\alpha_r)$.
In particular we have an isomorphism $$q_1\colon q_1^{-1}(M^i_-(\alpha_r))\cong M^i_-(\alpha_r).$$
\item[(2)] The morphism $q_2\colon M_-(\alpha_r,i)\to M_-(\alpha_{r-i})$ is a $Gr(n+1-r+i,i)$-bundle. In particular, we have an isomorphism $q_2\colon M_-(\alpha_{n+1},i)\cong  M_-(\alpha_{n+1-i})$.
\item[(3)] For any $j\ge 0$, we have
$q_1^{-1}(M_-^{i+j}(\alpha_r))\cong
q_2^{-1}(M_-^{j}(\alpha_{r-i}))$.
\item[(4)] The morphism $q'_1\colon M_+(\alpha_r,i)\to M_+(\alpha_r)$ is a $Gr(j,i)$-bundle over each strata $M_+^j(\alpha_r)$.
In particular we have an isomorphism $$(q'_1)^{-1}(M_+^j(\alpha_r))\cong M_+^j(\alpha_r).$$
\item[(5)] The morphism $q'_2\colon M_+(\alpha_r,i)\to M_+(\alpha_{r-i})$ is a $Gr(n-2-r+i,i)$-bundle. In particular we have an isomorphism $q'_2\colon M_+(\alpha_{n-2},i)\cong M_+(\alpha_{n-2-i})$.
\item[(6)] For any $j\ge 0$, we have
${q_1'}^{-1}(M_+^{i+j}(\alpha_r))\cong
{q_2'}^{-1}(M_+^{j}(\alpha_{r-i}))$.
\end{itemize}
\end{prop}
\begin{proof}
(1)  The fiber of $q_1$ over $E_-\in M^j_-(\alpha_r)$ is parametrized by $Gr(\Hom_B(S_0,E_-),i)$ for all $j\ge i$.
For the universal bundle $\U$ in (\ref{univ1}), as in \S~\ref{subsec:3-3} we put $\delta^{\ast}:=\sum_i\delta^{\ast}_i \otimes x_i\colon\U_0\to\U_1\otimes\mathbf x$.
Then for any point $p\in M_-(\alpha_r)$, we have $\Hom_B(S_0,\U_p)\cong (\ker \delta^{\ast})_p$.
Since $\ker \delta^{\ast}$ is locally free of rank $j$ on $M^j_-(\alpha_r)$ (cf. \cite[Chapter II]{ACGH}), we have $Gr(j,i)$-bundle $Gr(\ker \delta^{\ast}|_{M_-^j(\alpha_r)},i)$ on $M^j_-(\alpha_r)$.

On the other hand, by the definition of
$M^j_-(\alpha_r)$ (cf \S~\ref{subsec:3-2}),
we easily see that $M^j_-(\alpha_r)$
represents the moduli functor
parametrizing families of $\theta_-$-semistable
$B$-modules $E_-$ with $[E_-]=\alpha_r$ and
$\dim_{\C}\Hom_B(S_0,E_-)=j$.
Hence $q_1^{-1}(M^j_-(\alpha_r))$ have the same
universal property of
$Gr(\ker \gamma^{\ast}|_{M_-^j(\alpha_r)},i)$ and
we have $q_1^{-1}(M^j_-(\alpha_r))
\cong Gr(\ker \gamma^{\ast}|_{M^j_-(\alpha_r)},i)$.\\
(2) The fiber of $q_2$ over $E'=q_2(\bar{E}_-)$
is parametrized by $Gr\left(\Ext^1_B(E',S_0),i\right)$.
For the universal family
$\U'=\left(\U'_{-1}\stackrel{{\gamma'}^{\ast}_i}
{\to}\U'_0\stackrel{{\delta'}^{\ast}_j}{\to}\U'_1\right)$
of $B$-modules on $M_-(\alpha_{r-i})$,
we put $${\gamma'}^{\ast}:=\sum_i
{\gamma'}^{\ast}_i\otimes x^{\ast}_i\colon
\U_{-1}\otimes\mathbf x\to\U_0.$$
Since we have $(\ker{\gamma'}^{\ast})^{\vee}_{p'}
\cong \Ext^1_B(\U'_{p'},S_0)$ for any
$p'\in M_-(\alpha_{r-i})$.
Similarly as in (1) we get $$M_-(\alpha_r,i)\cong
Gr\left((\ker{\gamma'}^{\ast})^{\vee},i\right).$$\\
(3) Since spaces of both sides have the same
universal property, our claim holds.\\
(4), (5) and (6) are proved similarly as in
(1), (2) and (3).
\end{proof}

\begin{coro}
$M^i_-(\alpha_r)$ and $M^i_+(\alpha_r)$
are smooth for any $i,r\ge 0$.
\end{coro}
\begin{proof}
The restriction of the morphism $q_1\colon M_-(\alpha_r,i)\to M_-(\alpha_r)$ gives an isomorphism $$q_1^{-1}(M_-^i(\alpha_r))\cong M_-^i(\alpha_r).$$
By Proposition~\ref{grbun}~(3) we have an isomorphism $M_-^i(\alpha_r)\cong q_2^{-1}(M_-^0(\alpha_{r-i}))$.
Hence by Proposition~\ref{grbun}~(2), $M_-^i(\alpha_r)$ is isomorphic to a Grassmann-bundle over $M_-^0(\alpha_{r-i})$.
Since $M_-^0(\alpha_{r-i})$ is smooth by Proposition~\ref{sm}, we see that $M_-^i(\alpha_r)$ is also smooth.
Similarly $M_+^i(\alpha_r)$ is shown to be smooth.
\end{proof}

\subsection{Stratified Grassmann bundle}
\label{subsec:3-6}
In this section we show that morphisms $f_{\pm}\colon M_{\pm}(\alpha_r)\to M_0(\alpha_r)$ are described by stratified Grassmann bundles using Proposition~\ref{grbun}.

We consider the diagram: $$\xymatrix{&\ar[dl]_{\cong}^{q_2}M_-(\alpha_{n+1},n+1-r)\ar[dr]_{q_1}&\\M_-(\alpha_{r})&&M_-(\alpha_{n+1}).
}$$
By Proposition~\ref{grbun}~(2), $q_2$ is an isomorphism and we have a map $q_1\circ q_2^{-1}\colon M_-(\alpha_r)\to M_-(\alpha_{n+1})$,
which coincides with $g_-$ by (\ref{ex2}).
This gives another proof of Theorem~\ref{yos}.
Similarly the map $q_1'\circ{q_2'}^{-1}\colon M_+(\alpha_r)\cong M_+(\alpha_{n-2},n-2-r)\to M_+(\alpha_{n-2})$ coincide with the map $g_+$ by (\ref{ex3}).
For any $r\ge 0$, we have isomorphisms $M^0_-(\alpha_r)\cong M^{n+1-r}_-(\alpha_{n+1})$ and $M^0_+(\alpha_r)\cong M^{n-2-r}_+(\alpha_{n-2})$ via $g_-$ and $g_+$ respectively.
In particular, we have isomorphisms $$M^0_-(\alpha_{r-i})\cong M^{n+1-r+i}_-(\alpha_{n+1}), M^0_+(\alpha_{r-i})\cong M^{n-2-r+i}_+(\alpha_{n-2}).$$
By the diagram~(\ref{kro}), $M^{n+1-r+i}_-(\alpha_{n+1})$ and $M^{n-2-r+i}_-(\alpha_{n-2})$ coincide with images $f_-(M_-^i(\alpha_r))\cong f_+(M_+^i(\alpha_r))$ of $f_-$ and $f_+$.
This gives a proof of (2) in Main Theorem~\ref{inmain}.

By Proposition~\ref{grbun} and the diagram~(\ref{kro}) we also have proofs of (3) and (4) in Main Theorem~\ref{inmain}.

\subsection{Hodge polynomials of flips}
\label{subsec:3-7}
We study the difference between Hodge polynomials of $M_-(\alpha_r)$ and $M_+(\alpha_r)$.
To do this we use the virtual Hodge polynomial $e(Y):=\sum_{p,q}e^{p,q}(Y)x^py^q$ for any variety $Y$ (cf. \cite{DK}).

By Main Thoerem~\ref{inmain} we get the following diagram
\begin{equation}
\xymatrix{
\sqcup M_+^i(\alpha_r)\ar[rd]^{f_+}& &\sqcup M_-^i(\alpha_r) \ar[ld]_{f_-}\\
&\sqcup M_0^i(\alpha_r)&}
\end{equation}
where restrictions of $f_+$ and $f_-$ to $M_\pm^i$ are $Gr(n-2-r+i,i)$-bundle and $Gr(n+1-r+i,i)$-bundle over $M^i_0(\alpha_r)\cong M_\pm^0(\alpha_{r-i})$, respectively.
Hence we get the following equality.
\begin{multline}\label{diff}
e\left(M_-(\alpha_r)\right)-e\left(M_+(\alpha_r)\right)=\\
\sum_{i>0}\Big(e\left(Gr(n+1-r+i,i)\right)-e\left(Gr(n-2-r+i,i)\right)\Big)e\left(M_-^0(\alpha_{r-i})\right).
\end{multline}

In the following we compute the Hodge polynomial of $M_+(\alpha_r)$ from that of $M_-(\alpha_r)$ in the case where $r=1,2$.
In this case, we know the Hodge polynomial of $M_-(\alpha_r)\cong M_{\PP^2}(r,1,n)$ from \cite{ES} and \cite{Y1}.
We need the following proposition.
\begin{prop}\label{p2}
We have following isomorphisms:
$$M_-(\alpha_0)\cong M_+(\alpha_0)\cong\PP^2.$$
\end{prop}
A proof of this proposition is given in Appendix.
From this proposition and (\ref{diff}), we get the following: $$\cdots=M_+(1,1,1)=M_+(1,1,2)=\emptyset,$$
\begin{equation*}
\begin{split}
e\left(M_+(1,1,3)\right)&=t^{12}+t^{10}+3t^8+3t^6+3t^4+t^2+1,\\
e\left(M_+(1,1,4)\right)&=t^{16}+2t^{14}+5t^{12}+8t^{10}+10t^8+8t^6+5t^4+2t^2+1,\\
e\left(M_+(1,1,5)\right)&=\cdots+21t^{10}+19t^8+11t^6+6t^4+2t^2+1,\\
e\left(M_+(1,1,n)\right)&=e\left(M_{\PP^2}(1,1,n)\right)-
(t^{2n+4}+2t^{2n+2}+3t^{2n}+2t^{2n-2}+t^{2n-4}),
\end{split}
\end{equation*}
and $$\cdots=M_+(2,1,1)=M_+(2,1,2)=M_+(2,1,3)=\emptyset,$$
\begin{equation*}
\begin{split}
e\left(M_+(2,1,4)\right)=\cdots&+12t^{12}+10t^{10}+8t^8+5t^6+3t^4+t^2+1,\\
e\left(M_+(2,1,5)\right)=\cdots&+67t^{16}+60t^{14}+48t^{12}+32t^{10}+20t^8+10t^6\\
&+5t^4+2t^2+1,
\end{split}
\end{equation*}
where $t=xy$.

redpajama_set_name: RedPajamaArXiv