Toronto, ON – Follow Me Foodie to Kensington Market!
Follow Me Foodie to Kensington Market!
As a "foodie" it was one of my favourite parts of town to explore in Follow Me Foodie to Toronto! Kensington Market is just one component of the Kensington neighbourhood, but it is the heart of it. It is one of the oldest and most multicultural districts in downtown Toronto.
The area is now an eclectic mix of old and new businesses and the market is full of iconic institutions selling meats, cheeses, seafood, spices and produce.
Being from Vancouver, BC where Latin eateries and specialty Latin grocery stores are severely limited, I found that part alone worth the visit.
The area has almost some of everything. It is culturally satisfying and inspiring especially for those with a culinary curiosity. It's a multicultural and hippie area frequented by locals, but it is great for tourists looking for an unique and ethnic experience. It would be a shame to visit Toronto and miss out on this. The city prides itself on diversity and it feels celebrated here.
Over the last few years many of the older businesses have retired and new business that are more stylized or upscale are taking over. The area still has charm and character, but I did see and feel the "newcomers", although many are still independently owned and operated. There was a hippie–hipster vibe to it, but it wasn't pretentious and still very approachable. I could feel it was in the process of a makeover, but I hope it doesn't lose its rawness and authenticity. It is what makes it special.
The neighbourhood used to be predominantly Jewish, but after they moved to more wealthier areas of Toronto, new immigrants moved in. These immigrants were from the Caribbean or East Asia and many have stayed or settled in Chinatown which is only east of Kensington. I didn't get to explore the entire area as thoroughly as I would have liked, but I got a feel for it and I was infatuated.
There was one point when I was hesitant because the streets are narrow, a lot of the buildings look run down, and it can feel quite shady. It was broad daylight and the area is safe, but when I suddenly heard "As I walk through the valley of the shadow of death I take a look at my life and realize there's nothin' left…" ("Gangsta's Paradise" by Coolio) blaring from a boom box with a heavy base, I couldn't help but to question the situation. Was I still in Toronto? Yes, very much so. I got over it quickly, but it really made for an ultimate Kensington Market experience. It was almost like it was meant to happen and it added a strange, but very appropriate auditory component to my whole experience.
I had just come from lunch at Banh Mi Boys and Arepa Cafe, so I wasn't necessarily hungry… but who are we kidding? When has that ever really stopped me? Especially when I'm traveling! You make room because you don't know when you'll be back. So where was my first stop in Kensington Market?
Patty King! I don't really know why, but it felt authentic and legit and it looked like one of the older places in the market. It looked like an original and I had to try something that has always been there.
George! Hmm. Not exactly who I was expecting to be running this Jamaican patty shop; although Jamaica does have a Chinese population, and Jamaican patties were invented by a Chinese-Jamaican family. Anyhow, George is the owner and operator of Patty Shop and yes, he has the full Jamaican accent. It was a legit "yeah man" and not a "yeah mon". I started talking to him about Follow Me Foodie to Jamaica and it is where he was originally from.
I ended up ordering a Curry Goat Pattie ($1.45) and a Sweet Potato Pudding ($1.99). I wanted something sweet because I hadn't had dessert yet and I saw two guys chomping away at these. Sweet Potato Pudding is a popular Jamaican dessert made from sweet potatoes, flour, sugar, coconut milk, and warm spices like cinnamon and nutmeg. It's very dense and a bit rubbery. It's a bit acquired and not really my thing yet, but I doubt this is the best it gets and I just have to find some way to get my hands on a "Jamaican grandmother's home made version".
The Goat Curry Pattie was the recommended pattie although not as popular with those unfamiliar with Jamaican food or goat. A few Jamaicans who frequent the place recommended it to me, so it was an easy decision. There wasn't much filling it in, but that's how they are in Jamaica too – see my posts for Juici Patties and Tastee Patties which are the two most popular and famous Jamaican patty places in Jamaica. The Goat Curry Pattie was okay, but perhaps I'd feel more passionate about it if I were a local.
Next stop! Thomas Lavers Cannery & Delicatessen. I was so happy to come across this. It was on my original Follow Me Foodie to Toronto itinerary, but every Toronto "foodie" I mentioned it to had no idea what I was talking about.
Really?! Have I, a tourist, found something locals don't even know about yet?! Yes, there's a childish satisfaction in that. I didn't even know it was located in Kensington Market, but it was and I found it unexpectedly. It was fate.
Yup, it was definitely in tune with the Top 10 Food Trends of 2013 and the only reason locals don't know about it yet is because it only opened December 2012. The owners are restaurateurs Brian Lavers and Tye Thomas, and Thomas Lavers Cannery & Delicatessen is supposed to be a revamped ode to the area.
They specialize in house cured, brined, smoked, and roasted deli meats, hand crafted fresh pastas, home made ready-to-go meals, house made artisan sodas, and a variety of pickles and preserves made from local Ontario produce where possible. If you didn't notice the theme, everything is made in house. They also have vegan options to cater to those in the area, so it's a nice neighbourhood gem.
They welcome you to sample their preserves before purchasing too. They offer smoked ketchup, house mustard, tomato butter, mushroom walnut pâté and many other condiments. The kimchi jam was one I particularly liked and they make great gifts ranging from about $5-8.
It's not a restaurant, but a grab n' go place and their deli sandwiches are their ready to eat feature. All condiments are made in house and you can choose from: roast beef, corned beef, roast turkey, smoked ham, Muffuletta, vegan reuben, and a vegan banh mi. Meat sandwiches $8-9 and vegan sandwiches are $7. Get one alongside a house made artisan soda – root beer or ginger beer ($3).
Did you catch it?! I'm not sure if you caught it in the last picture, but there's no way I missed it.
Here it is again. The same picture. Do you see it?
There's no missing it now right?!
The Grilled Cheese. How could I not?!
I know it's not "ethnic", but it's nostalgic food and it was cold outside – nothing like cheese to warm you up. This is a sit down restaurant and it's one of the newer, hip, "next generation" places that opened in the Kensington neighbourhood.
I know. Is it really necessary to have a sit down restaurant serving just grilled cheese sandwiches?! Isn't that something we can all make at home? Yes, but it's also important to note that it opened right across the street from a pub. Great business opportunity. Late night grilled cheese sandwiches anyone? Sing me up. It is classic comfort food, and every non-lactose intolerant person has a soft spot for it. I even doubt you cheese haters… try it in a sandwich.
I ordered the Mushroom Madness Grilled Cheese Sandwich – Cheddar, Swiss, Portobello and Cremini Mushrooms ($6) with added egg (+$2) and added avocado (+$2). I would have added these items if I didn't come from Banh Mi Boys, Arepa Cafe, Patty King, Thomas Lavers Cannery & Delicatessen… and basically 4 lunches. It came with a side of chips which tasted like regular Ruffles chips and a pickle. I think they should eventually source the pickles from Thomas Lavers especially since the area feels so community based.
Are you convinced yet? A grilled cheese is a grilled cheese, but this was still a very good and affordable one. For Vancouverites, Mom's Grilled Cheese wins the grilled cheese match, but I wouldn't say no to this.
… and then I came back the next morning to try it.
Nu Bügel opened late January this year and just like Thomas Lavers it's a throwback shop. It is an ode to Jewish tradition being brought back to Kensington Market in a modern and exciting way.
In North America the bagel rivalry is between the Montreal style bagel and the New York style bagel, although European and Asian style bagels exist. Well to be more exact, the bagel was invented in Poland (some argue Austria or Germany) and it was later introduced to America by a Jewish immigrant. The recipe was slightly altered according to area, but there are many different styles of bagels. Nu Bügel follows the traditions of a Montreal style bagel (boiled in honey water and baked in a wood fired oven), but the name bügel is Austrian-German. I won't get into German-Polish history because I don't know enough about it, but I'll thank them both for inventing the bügel.
I ordered one of each ($1 each/$5 half dozen/$9 dozen), but sesame is the "purist" flavour followed by poppyseed and that's how I usually "judge" bagels. As a tourist in Toronto for a limited time I wasn't going to search the city for "the best bagel" although this article highlights all the places for one.
I've done the St-Viateur Bagel & Café VS Fairmount Bagel in Follow Me Foodie to Montreal where bagels are actually their specialty, so I didn't feel it was necessary to do in Toronto. These ones were not as sweet as the Montreal style bagels from Fairmount which I personally like best, but they were still very good. They were served warm, a bit dense and chewy with a moist interior and a crisp exterior.
As a tourist I wouldn't say it is a "must try" if you've had the Montreal bagels from the bagel institutions in Montreal, but if I were a local I would come back to Nu Bügel. It's already a local favourite with line ups in the morning and it's a casual breakfast spot with affordable prices.
Everything bagel with cream cheese ($2).
Love the Kensington Market post! Glad you enjoyed our city. Come back during warmer weather and you can visit Pedestrian Sunday. They shut down the roads in Kensington, making it fully pedestrian-friendly, and they have street performers, and outdoor food vendors.
Kensington Market is an awesome place for foodies. The next time you're back, I heavily recommend Rasta Pasta (a new Jamaican place), Mexican Salsas Grill (Mexican), Emporium Latino (Salvadorean grocery store but they have a kitchen at the back – their pupusas seriously rock), Thai Noodles, and Segovia Meats for various South American dishes (mostly Chilean but also Colombian and even Brazilian). And close by on Dundas, very close to Spadina is a 24 hour Vietnamese place called Pho Pasteur that has some seriously good pho (although I don't recommend anything else there!).
Hungry to help the Downtown Eastside? Sometimes the heart of a restaurant goes beyond its food. Ambiance? Good. Service? Attentive. Value? Affordable. Food? Great. Now where is the line about corporate social responsibility? The restaurant world used to be about bringing good food to the table, but tables are turning and social change is important.


To answer these questions I use data from a variety of telescopes, including the latest all-sky surveys. I also have an active interest in the international Virtual Observatory effort, which is developing standards for the discovery, exchange and analysis of astronomical data. I completed my Ph.D. at the Australian National University's Mount Stromlo Observatory in 2012.
…for a more technical description of my work, see my research page.
…more on my publications page.
Niall Deacon (Hertfordshire), Josh Schlieder (NASA Ames) and I recently discovered that two previously known young objects, the M dwarf TYC 9486-927-1 and the L3 brown dwarf 2MASS J2126-8140, are in fact co-distant (32 pc), co-moving and form a 6900 AU wide pair. Using a new spectroscopic 10–45 Myr age for TYC 9486, we calculate that 2MASS J2126 is a gas giant planet or low-mass brown dwarf of 11–15 Jupiter masses. This makes it by far the widest known planetary mass companion and a much wider analogue to the well-studied exoplanet Beta Pictoris b.


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\def\noprint{\noprint}

\title[Semi-projective toric DM stacks]{Note on Orbifold Chow Ring of Semi-projective Toric Deligne-Mumford Stacks}

\author{Yunfeng Jiang}
\address{Department of Mathematics\\ University of British Columbia\\ 1984 Mathematics Road\\Vancouver\\ BC V6T 1Z2\\ Canada}
\email{jiangyf@math.ubc.ca}

\author{Hsian-Hua Tseng}
\address{Department of Mathematics\\ University of British Columbia\\ 1984 Mathematics Road\\Vancouver\\ BC V6T 1Z2\\ Canada}
\email{hhtseng@math.ubc.ca}

\date{\today}

\begin{document}
\begin{abstract}
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
\end{abstract}

\maketitle

\section{Introduction}

The main goal of this paper is to generalize the orbifold Chow ring formula of Borisov-Chen-Smith for
projective toric DM stacks to the case of semi-projective toric DM stacks.

In the paper \cite {BCS}, Borisov, Chen, and Smith developed the theory of toric DM stacks using stacky fans. A stacky fan is a triple $\mathbf{\Sigma}=(N,\Sigma,\beta)$, where $N$ is a finitely generated abelian group, $\Sigma$ is a simplicial fan in the lattice $\overline{N}:=N\slash \text{torsion}$ and $\beta: \mathbb{Z}^n\rightarrow N$ is a map given by a collection of vectors $\{b_1,\cdots,b_n\}\subset N$ such that the images $\{\overline{b}_{1},\cdots,\overline{b}_{n}\}$ generate the fan $\Sigma$.  A toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ is defined using $\mathbf{\Sigma}$; it is a quotient stack whose coarse moduli space is the toric variety $X(\Sigma)$ corresponding to the simplicial fan $\Sigma$.

The construction of toric DM stacks was slightly generalized later in \cite{Jiang}, in which the notion of extended stacky fans was introduced. This new notion is based on that of stacky fans plus some extra data. Extended stacky fans yield toric DM stacks in the same way as stacky fans do. The main point is that extended stacky fans provide presentations of toric DM stacks not available from stacky fans.

When $X(\Sigma)$ is projective, it is found in \cite{BCS} that the orbifold Chow ring (or Chen-Ruan cohomology ring) of $\mathcal{X}(\mathbf{\Sigma})$ is isomorphic to a deformed ring of the group ring of $N$. We call a toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ {\em semi-projective} if its coarse moduli space $X(\Sigma)$ is semi-projective. Hausel and Sturmfels \cite{HS} computed the Chow ring of semi-projective toric varieties. Their answer is also known as the ``Stanley--Reisner'' ring of a fan. Using their result, we prove a formula of the orbifold Chow ring of semi-projective toric DM stacks.

Consider an extended stacky fan $\mathbf{\Sigma}=(N,\Sigma,\beta)$, where $\Sigma$ is the simplicial fan of the semi-projective  toric variety $X(\Sigma)$. Let $N_{tor}$ be the torsion subgroup of $N$, then $N=\overline{N}\oplus N_{tor}$. Let  $N_{\Sigma}:=|\Sigma|\oplus N_{tor}$. Note that $|\Sigma|$ is convex, so $|\Sigma|\oplus N_{tor}$ is a subgroup of $N$. Define the deformed ring $\mathbb{Q}[N_{\Sigma}]:=\bigoplus_{c\in N_{\Sigma}}\mathbb{Q}y^{c}$ with the product structure given by
\begin{equation}\label{productA}
y^{c_{1}}\cdot y^{c_{2}}:=\begin{cases}y^{c_{1}+c_{2}}&\text{if
there is a cone}~ \sigma\in\Sigma ~\text{such that}~ \overline{c}_{1}\in\sigma, \overline{c}_{2}\in\sigma\,;\\
0&\text{otherwise}\,.\end{cases}
\end{equation}
Note that if $\mathcal{X}(\mathbf{\Sigma})$ is projective, then $N_{\Sigma}=N$ and $\mathbb{Q}[N_{\Sigma}]$ is the deformed ring
$\mathbb{Q}[N]^{\mathbf{\Sigma}}$ in \cite{BCS}.
Let $A^{*}_{orb}(\mathcal{X}(\mathbf{\Sigma}))$ denote the orbifold Chow ring of the toric DM stack $\mathcal{X}(\mathbf{\Sigma})$.

\begin{thm}\label{main}
Assume that $\mathcal{X}(\mathbf{\Sigma})$ is semi-projective. There is an isomorphism of rings
$$A^{*}_{orb}(\mathcal{X}(\mathbf{\Sigma}))\cong \frac{\mathbb{Q}[N_{\Sigma}]}{\{\sum_{i=1}^{n}e(b_{i})y^{b_{i}}:e\in N^{\star}\}}.$$
\end{thm}

The strategy of proving Theorem \ref{main} is as follows. We use a formula in \cite{HS} for the ordinary Chow ring of semi-projective toric varieties. We prove that each twisted sector is also a semi-projective toric DM stack. With this, we use a method similar to that in \cite{BCS} and \cite{Jiang} to prove the isomorphism as modules. The argument to show the isomorphism as rings is the same as that in \cite{BCS}, except that we only take elements in the support of the fan.

An interesting class of examples of semi-projective toric DM stack is the Lawrence toric DM stacks. We discuss the properties of such stacks. We prove that each 3-twisted sector or twisted sector is again a Lawrence toric DM stack. This allows us to draw connections to hypertoric DM stacks studied in \cite{JT}. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack. This is an analog of Theorem 1.1 in \cite{HS} for orbifold Chow rings.

The rest of this text is organized as follows. In Section \ref{semi-pro} we define semi-projective toric DM stacks and prove Theorem 1.1. Results on Lawrence toric DM stacks are discussed in Section \ref{Lawrence}.
\subsection*{Conventions}
In this paper we work entirely algebraically over the field of
complex numbers. Chow rings and orbifold Chow rings are taken with
rational coefficients. By an orbifold we mean a smooth
Deligne-Mumford stack with trivial generic stabilizer.

For a simplicial fan $\Sigma$, we use $|\Sigma|$ to represent the lattice points in $\Sigma$.
Note that if $\Sigma$ is convex, $|\Sigma|$ is a free abelian subgroup of $N$.
We write $N^{\star}$ for $Hom_{\mathbb{Z}}(N,\mathbb{Z})$ and $N\to \overline{N}$ the natural map of modding out torsions. We refer to \cite{BCS} for the construction of the Gale dual
$\beta^{\vee}: \mathbb{Z}^{m}\to DG(\beta)$ of $\beta: \mathbb{Z}^{m}\to N$.

\subsection*{Acknowledgments}
We thank Kai Behrend and Nicholas Proudfoot for valuable discussions.
\section{Semi-projective toric DM stacks and their orbifold Chow rings}\label{semi-pro}
In this section we define semi-projective toric DM stacks and discuss their properties.

\subsection{Semi-projective toric DM stacks}

\begin{defn}[\cite{HS}]
A  toric variety $X$ is called semi-projevtive if the natural map
$$\pi: X\rightarrow X_{0}=\text{Spec}(H^{0}(X,\mathcal{O}_{X})),$$
is projective and $X$ has at least one torus-fixed point.
\end{defn}

\begin{defn}[\cite{Jiang}]\label{stackyfan}
An extended stacky fan $\mathbf{\Sigma}$ is a triple $(N,\Sigma,\beta)$,
where  $N$ is a finitely generated abelian group, $\Sigma$ is a
simplicial fan  in $N_{\mathbb{R}}$ and $\beta:
\mathbb{Z}^{m}\to N$ is the map determined by the
elements $\{b_{1},\cdots,b_{m}\}$ in $N$ such that
$\{\overline{b}_{1},\cdots,\overline{b}_{n}\}$ generate the
simplicial fan $\Sigma$ (here $m\geq n$).
\end{defn}
Given an extended stacky fan $\mathbf{\Sigma}=(N,\Sigma,\beta)$,
we have the following exact sequences:
\begin{equation}\label{exact1}
0\longrightarrow DG(\beta)^{\star}\longrightarrow
\mathbb{Z}^{m}\stackrel{\beta}{\longrightarrow} N\longrightarrow
Coker(\beta)\longrightarrow 0,
\end{equation}
\begin{equation}\label{exact2}
0\longrightarrow N^{\star}\longrightarrow
\mathbb{Z}^{m}\stackrel{\beta^{\vee}}{\longrightarrow}
DG(\beta)\longrightarrow Coker(\beta^{\vee})\longrightarrow 0,
\end{equation}
where $\beta^{\vee}$ is the Gale dual of $\beta$ (see \cite{BCS}). Applying $Hom_\mathbb{Z}(-,\mathbb{C}^{*})$ to (\ref{exact2}) yields
\begin{equation}\label{exact3}
1\longrightarrow \mu\longrightarrow
G\stackrel{\alpha}{\longrightarrow}
(\mathbb{C}^{*})^{m}\longrightarrow
(\mathbb{C}^{*})^{d}\longrightarrow 1. \
\end{equation}
The toric
DM stack $\mathcal{X}(\mathbf{\Sigma})$ is the quotient stack $[Z/G]$, where
$Z:=(\mathbb{C}^{n}\setminus V(J_{\Sigma}))\times (\mathbb{C}^{*})^{m-n}$, $J_{\Sigma}$ is the
irrelevant ideal of the fan $\Sigma$ and $G$ acts on $Z$ through the map $\alpha$ in (\ref{exact3}). The
coarse moduli space of $\mathcal{X}(\mathbf{\Sigma})$ is the simplicial toric variety
$X(\Sigma)$ corresponding to the simplicial fan $\Sigma$, see \cite{BCS} and \cite{Jiang}.

\begin{defn}\label{semi-toric}
A toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ is {\em semi-projective} if the coarse
moduli space $X(\Sigma)$ is semi-projective.
\end{defn}

\begin{thm}\label{semitor}
The following notions are equivalent:
\begin{enumerate}
\item A semi-projective toric DM stack $\mathcal{X}(\mathbf{\Sigma})$;
\item A toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ such that the simplicial fan $\Sigma$
is a regular triangulation of $\mathcal{B}=\{\overline{b}_{1},\cdots,\overline{b}_{n}\}$
which spans the lattice $\overline{N}$.
\end{enumerate}
\end{thm}

\begin{pf}
Since the toric DM stack is semi-projective if its coarse moduli space is semi-projective,
the theorem follows from results in \cite{HS}.
\end{pf}

\subsection{The inertia stack}
Let $\mathbf{\Sigma}$ be an extended stacky fan and $\sigma\in\Sigma$ a cone. Define $link(\sigma):=\{\tau:\sigma+\tau\in \Sigma, \sigma\cap \tau=0\}$. Let $\{\widetilde{\rho}_{1},\ldots,\widetilde{\rho}_{l}\}$ be the rays in $link(\sigma)$. Consider the quotient extended stacky fan $\mathbf{\Sigma/\sigma}=(N(\sigma),\Sigma/\sigma,\beta(\sigma))$, with $\beta(\sigma): \mathbb{Z}^{l+m-n}\to N(\sigma)$ given by the images of $b_{1},\ldots,b_{l}$ and $b_{n+1},\ldots,b_{m}$
under $N\to N(\sigma)$. By the construction of toric Deligne-Mumford stacks,
if $\sigma$ is contained in a top dimensional cone in $\Sigma$, we have $\mathcal{X}(\mathbf{\Sigma/\sigma}):=[Z(\sigma)/G(\sigma)]$,
where $Z(\sigma)=(\mathbb{A}^{l}\setminus\mathbb{V}(J_{\Sigma/\sigma}))\times (\mathbb{C}^{*})^{m-n}$ and $G(\sigma)=Hom_{\mathbb{Z}}(DG(\beta(\sigma)),\mathbb{C}^{*})$.

\begin{lem}
If $\mathcal{X}(\mathbf{\Sigma})$ is semi-projective, so is $\mathcal{X}(\mathbf{\Sigma/\sigma})$.
\end{lem}

\begin{pf}
Semi-projectivity of the stack $\mathcal{X}(\mathbf{\Sigma})$ means the simplicial fan $\Sigma$ is a fan coming from a regular triangulation of $\mathcal{B}=\{\overline{b}_{1},\cdots,\overline{b}_{n}\}$ which spans the lattice $\overline{N}$. Let $pos(\mathcal{B})$ be the convex polyhedral cone generated by $\mathcal{B}$. Then from \cite{HS}, the triangulation is supported on $pos(\mathcal{B})$ and is dermined by a simple polyhedron whose normal fan is $\Sigma$. So $\sigma$ is contained in a top-dimensional
cone $\tau$ in $\Sigma$. The image $\widetilde{\tau}$ of $\tau$ under quotient by $\sigma$ is a top-dimensional cone in the quotient fan $\Sigma/\sigma$. So the toric variety $X(\Sigma/\sigma)$ is semi-projective by Theorem \ref{semitor}, and the stack $\mathcal{X}(\mathbf{\Sigma/\sigma})$ is semi-projective by definition.
\end{pf}

Recall in \cite{BCS} that for each top-dimensional cone $\sigma$ in $\Sigma$, define $Box(\sigma)$ to be the set of elements $v\in N$ such that $\overline{v}=\sum_{\rho_{i}\subseteq \sigma}a_{i}\overline{b}_{i}$ for some $0\leq a_{i}<1$. Elements in  $Box(\sigma)$ are in one-to-one correspondence with elements in the finite group $N(\sigma)=N/N_{\sigma}$, where $N(\sigma)$ is a local group of the stack
$\mathcal{X}(\mathbf{\Sigma})$. In fact, we write $\overline{v}=\sum_{\rho_{i}\subseteq \sigma(\overline{v})}a_{i}\overline{b}_{i}$ for some $0<a_{i}<1$, where $\sigma(\overline{v})$ is the minimal cone containing $\overline{v}$. Denoted by $Box(\mathbf{\Sigma})$ the union  of $Box(\sigma)$ for all top-dimensional cones $\sigma$.

\begin{prop}
The $r$-inertia stack is given by
\begin{equation}\label{inertia}
\mathcal{I}_{r}\left(\mathcal{X}(\mathbf{\Sigma})\right)=\coprod_{(v_{1},\cdots,v_{r})\in Box(\mathbf{\Sigma})^{r}}
\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\cdots,\overline{v}_{r})),
\end{equation}
where $\sigma(\overline{v}_{1},\cdots,\overline{v}_{r})$ is the minimal cone in
$\Sigma$  containing $\overline{v}_{1},\cdots,\overline{v}_{r}$.
\end{prop}
\begin{pf}
Since $G$ is an abelian group, we have
$$\mathcal{I}_{r}\left(\mathcal{X}(\mathbf{\Sigma})\right)=[(\coprod_{(v_{1},\cdots,v_{r})\in (G)^{r}}
Z^{(v_{1},\cdots,v_{r})})\slash G],$$
where $Z^{(v_{1},\cdots,v_{r})}\subset Z$ is the subvariety fixed by $v_{1},\cdots,v_{r}$. Since
$\sigma(\overline{v}_{1},\cdots,\overline{v}_{r})$ is contained in a top-dimensional cone  in
$\Sigma$. We use the same method as in Lemma 4.6 and Proposition 4.7 of \cite{BCS} to prove that
$[Z^{(v_{1},\cdots,v_{r})}\slash G]\cong \mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\cdots,\overline{v}_{r}))$.
\end{pf}

Note that in (\ref{inertia}) each component is semi-projective.

\subsection{The orbifold Chow ring}\label{ring}
In this section we compute the orbifold Chow ring of semi-projective toric DM stacks and prove Theorem \ref{main}.

\subsubsection{The module structure}

Let $\mathbf{\Sigma}=(N,\Sigma,\beta)$ be an extended  stacky fan such that the toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ is semi-projective. Since the fan $\Sigma$ is convex, $|\Sigma|$ is an abelian subgroup of $N$. We put $N_{\Sigma}:=|\Sigma|\oplus N_{tor}$, where $N_{tor}$ is the torsion
subgroup of $N$.  Define the deformed ring $\mathbb{Q}[N_{\Sigma}]:=\bigoplus_{c\in N_{\Sigma}}\mathbb{Q}y^{c}$ with the product structure given by (\ref{productA}).

Let $\{\rho_{1},\ldots,\rho_{n}\}$ be the rays of $\Sigma$, then each $\rho_{i}$ corresponds to a line
bundle $L_{i}$ over the toric Deligne-Mumford stack $\mathcal{X}(\mathbf{\Sigma})$ given by the trivial line bundle $\mathbb{C}\times Z$ over $Z$ with the $G$ action on $\mathbb{C}$ given by the $i$-th component $\alpha_{i}$ of $\alpha: G\to (\mathbb{C}^{*})^{m}$ in (\ref{exact3}). The first Chern classes of the line bundles $L_{i}$, which we identify with $y^{b_{i}}$, generate the cohomology ring of the simplicial toric variety $X(\Sigma)$.

Let $S_{\mathbf{\Sigma}}$ be the quotient ring $\frac{\mathbb{Q}[y^{b_{1}},\cdots,y^{b_{n}}]}{I_{\Sigma}}$, where $I_{\Sigma}$ is the square-free ideal of the fan $\Sigma$ generated by the monomials
$$\{y^{b_{i_{1}}}\cdots y^{b_{i_{k}}}: \overline{b}_{i_{1}},\cdots, \overline{b}_{i_{k}} \text{ do not generate a cone in }\Sigma\}.$$
It is clear that $S_{\mathbf{\Sigma}}$ is a subring of the deformed ring $\mathbb{Q}[N_{\Sigma}]$.

\begin{lem}
Let $A^{*}(\mathcal{X}(\mathbf{\Sigma}))$ be the ordinary Chow ring of a semi-projective toric DM stack $\mathcal{X}(\mathbf{\Sigma})$. Then there is a ring isomorphism:
$$A^{*}(\mathcal{X}(\mathbf{\Sigma}))\cong \frac{S_{\mathbf{\Sigma}}}
{\{\sum_{i=1}^{n}e(b_{i})y^{b_{i}}: e\in N^{\star}\}}.$$
\end{lem}

\begin{pf}
The Lemma is easily proven from the fact that the Chow ring of a DM stack is isomorphic to the Chow ring of its coarse moduli space (\cite{V}) and Proposition 2.11 in \cite{HS}.
\end{pf}

Now we study the module structure on $A_{orb}^{*}\left(\mathcal{X}(\mathbf{\Sigma})\right)$. Because $\Sigma$ is a simplicial fan, we have:

\begin{lem}\label{smalllemma}
For any $c\in N_{\Sigma}$, let $\sigma$ be the minimal cone in $\Sigma$ containing $\overline{c}$. Then there is a unique expression $c=v+\sum_{\rho_{i}\subset\sigma}m_{i}b_{i}$ where $m_{i}\in \mathbb{Z}_{\geq 0}$, and $v\in Box(\sigma)$.
\end{lem}

\begin{prop}\label{vectorspace}
Let $\mathcal{X}(\mathbf{\Sigma})$ be a semi-projective toric  DM stack associated to an extended  stacky fan $\mathbf{\Sigma}$. We have an isomorphism of $A^{*}(\mathcal{X}(\mathbf{\Sigma}))$-modules:
$$\bigoplus_{v\in Box(\mathbf{\Sigma})}A^{*}\left(\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}))\right)[deg(y^{v})]\cong \frac{\mathbb{Q}[N_{\Sigma}]}{\{\sum_{i=1}^{n}e(b_{i})y^{b_{i}}: e\in N^{\star}\}}.$$
\end{prop}

\begin{pf}
From the definition of $\mathbb{Q}[N_{\Sigma}]$ and Lemma \ref{smalllemma}, we see  that $\mathbb{Q}[N_{\Sigma}]=\bigoplus_{v\in Box(\mathbf{\Sigma})}y^{v}\cdot S_{\mathbf{\Sigma}}$. The rest is similar to the proof of Proposition 4.7 in \cite{Jiang}, we leave it to the readers.
\end{pf}

\subsubsection{The Chen-Ruan product structure}

The orbifold cup product on a DM stack $\mathcal{X}$ is defined using genus zero, degree zero 3-pointed orbifold Gromov-Witten invariants on $\mathcal{X}$. The relevant moduli space is the disjoint union of all  3-twisted sectors (i.e. the double inertia stack). By (\ref{inertia}), the 3-twisted sectors of a semi-projective toric DM stack $\mathcal{X}(\mathbf{\Sigma})$ are
\begin{equation}\label{3-sector}
\coprod_{(v_{1},v_{2},v_{3})\in Box(\mathbf{\Sigma})^{3},
v_{1}v_{2}v_{3}=1}
~\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})).
\end{equation}

Let $ev_{i}:
\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))\to
\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{i}))$ be the evaluation maps. The obstruction bundle (see \cite{CR2}) $Ob_{(v_{1},v_{2},v_{3})}$ over the 3-twisted sector
$\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))$ are defined by
\begin{equation}\label{obstruction}
Ob_{(v_{1},v_{2},v_{3})}:=\left(e^{*}T\left(\mathcal{X}(\mathbf{\Sigma})\right)\otimes
H^{1}(C,\mathcal{O}_{C})\right)^{H},
\end{equation}
where $e: \mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))\to
\mathcal{X}(\mathbf{\Sigma})$ is the embedding,  $C\to \mathbb{P}^{1}$ is the $H$-covering branched over three marked points $\{0,1,\infty\}\subset \mathbb{P}^{1}$, and  $H$ is the group generated by $v_{1},v_{2},v_{3}$.

A general result in \cite{CH} and \cite{JKK} about the obstruction bundle implies the following.

\begin{prop}\label{obstructionbdle}
Let
$\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))$ be a 3-twisted sector of the stack $\mathcal{X}(\mathbf{\Sigma})$. Suppose $v_{1}+v_{2}+v_{3}=\sum_{\rho_{i}\subset
\sigma(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})}a_{i}b_{i}$, $a_{i}=1$ or $2$. Then the Euler class of the obstruction bundle $Ob_{(v_{1},v_{2},v_{3})}$ on
$\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))$
is
$$\prod_{a_{i}=2}c_{1}(L_{i})|_{\mathcal{X}(\mathbf{\Sigma/\sigma}(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3}))},$$
where $L_{i}$ is the line bundle over $\mathcal{X}(\mathbf{\Sigma})$ corresponding to the ray $\rho_{i}$.
\end{prop}

Let $v\in Box(\mathbf{\Sigma})$, say $v\in N(\sigma)$ for some top-dimensional cone $\sigma$. Let $\check{v}\in Box(\mathbf{\Sigma})$ be the inverse of $v$ as an element in the group $N(\sigma)$. Equivalently, if $v=\sum_{\rho_{i}\subseteq \sigma(\overline{v})}\alpha_{i}b_{i}$ for $0<\alpha_{i}<1$, then $\check{v}=\sum_{\rho_{i}\subseteq \sigma(\overline{v})}(1-\alpha_{i})b_{i}$. Then for $\alpha_{1},\alpha_{2}\in A^{*}_{orb}(\mathcal{X}(\mathbf{\Sigma}))$, the orbifold cup product is defined by
\begin{equation}\label{cupproduct}
\alpha_{1}\cup_{orb}\alpha_{2}=\widehat{ev}_{3*}(ev_{1}^{*}\alpha_{1}\cup ev_{2}^{*}\alpha_{2}\cup e(Ob_{(v_{1},v_{2},v_{3})})),
\end{equation}
where $\widehat{ev}_{3}=I\circ ev_{3}$, and $I: \mathcal{I}\mathcal{X}(\mathbf{\Sigma}) \rightarrow \mathcal{I}\mathcal{X}(\mathbf{\Sigma})$ is the natural map given by  $(x,g)\mapsto (x,g^{-1})$.

\subsubsection*{Proof of Theorem 1.1}
By Proposition \ref{vectorspace}, it remains to consider the cup product. In this case, for any
$v_{1},v_{2}\in Box(\mathbf{\Sigma})$, we also have
$$v_{1}+v_{2}=\check{v}_{3}+\sum_{a_{i}=2}b_{i}+\sum_{i\in J}b_{i},$$
where $J$ represents the set of $j$
such that $\rho_{j}$ belongs to
$\sigma(\overline{v}_{1},\overline{v}_{2})$, but not belong to
$\sigma(\overline{v}_{3})$. Then the proof is the same as the proof in \cite{BCS}.
We omit the details.

\section{Lawrence Toric DM stacks}\label{Lawrence}

In this section we study a special type of semi-projective toric DM stacks called the Lawrence toric DM stacks. Their orbifold Chow rings are shown to be isomorphic to the orbifold Chow rings of their associated hypertoric DM stacks studied in \cite{JT}.

\subsection{Stacky hyperplane arrangements}
Let $N$, $\{b_{1},\cdots,b_{m}\}\in N$, $\beta:\mathbb{Z}^{m}\to N$, and $\{\overline{b}_{1},\cdots,\overline{b}_{m}\}\subset \overline{N}$ be as in Definition \ref{stackyfan}.  We assume that $\{b_{1},\cdots,b_{m}\}\in N$ are nontorsion integral vectors. We still have the exact sequences (\ref{exact1}) and (\ref{exact2}). The Gale dual map $\beta^{\vee}$ of $\beta$ is given by a collection of integral vectors $\beta^{\vee}=(a_1,\cdots,a_m)$. Choose a generic element $\theta\in DG(\beta)$ and let $\psi:=(r_{1},\cdots,r_{m})$ be a lifting of $\theta$ in $\mathbb{Z}^{m}$ such that $\theta=-\beta^{\vee}\psi$. Note that $\theta$ is generic if and only if it is not in any hyperplane of the configuration determined by $\beta^{\vee}$ in $DG(\beta)_{\mathbb{R}}$. Associated to $\theta$ there is a hyperplane arrangement $\mathcal{H}=\{H_{1},\cdots,H_{m}\}$ defined as follows:  let $H_{i}$ be the hyperplane
\begin{equation}\label{arrangement}
H_{i}:=\{v\in M_{\mathbb{R}}|<\overline{b}_{i},v>+r_{i}=0\}\subset M_{\mathbb{R}}.
\end{equation}
This determines hyperplane arrangement in $M_{\mathbb{R}}$, up to translation.
It is well-known that hyperplane arrangements determine the topology of hypertoric varieties (\cite{BD}).
We call $\mathcal{A}:=(N,\beta,\theta)$ a {\em stacky hyperplane arrangement}.

The toric variety $X(\Sigma)$ is defined by the weighted polytope $\mathbf{\Gamma}:=\bigcap_{i=1}^{m}F_{i}$, where $F_{i}=\{v\in M_{\mathbb{R}}|<b_{i},v>+r_{i}\geq 0\}$. Suppose that $\mathbf{\Gamma}$ is bounded, the  fan $\Sigma$ is the normal fan of  $\mathbf{\Gamma}$ in $M_{\mathbb{R}}=\mathbb{R}^{d}$ with one dimensional rays generated by $\overline{b}_{1},\cdots,\overline{b}_{n}$. By reordering, we may assume that $H_{1},\cdots,H_{n}$ are the hyperplanes that bound the polytope $\mathbf{\Gamma}$, and $H_{n+1},\cdots,H_{m}$ are the other hyperplanes. Then we have an extended stacky fan
$\mathbf{\Sigma}=(N,\Sigma,\beta)$ as in Definition \ref{stackyfan}, with $\Sigma$ the normal fan of $\mathbf{\Gamma}$, $\beta:\mathbb{Z}^{m}\to N$ given by $\{b_{1},\cdots,b_{n},b_{n+1},\cdots,b_{m}\}\subset N$, and $\{b_{n+1},\cdots,b_{m}\}$ the extra data. We define the hypertoric DM stack $\mathcal{M}(\mathcal{A})$ using this $\mathcal{A}$, see \cite{JT} for more details.

\subsection{Lawrence toric DM stacks}
Applying Gale dual to the map
\begin{equation}\label{betaL}
\mathbb{Z}^{m}\oplus \mathbb{Z}^{m}\to DG(\beta),
\end{equation}
given by $(\beta^{\vee},-\beta^{\vee})$, we obtain
$$\beta_{L}: \mathbb{Z}^{m}\oplus \mathbb{Z}^{m}\longrightarrow N_{L},$$
which  is given by integral  vectors $\{b_{L,1},\cdots,b_{L,m},b'_{L,1},\cdots,b'_{L,m}\}$ in $N_{L}$. The natural images  $\{\overline{b}_{L,1},\cdots,\overline{b}_{L,m},\overline{b}'_{L,1},\cdots,\overline{b}'_{L,m}\}\subset \overline{N}_{L}$
are called the Lawrence lifting of $\{\overline{b}_{1},\cdots,\overline{b}_{m}\}\subset \overline{N}$.

Associated to the generic element $\theta$, let $\overline{\theta}$ be the natural image under the map $DG(\beta)\rightarrow \overline{DG(\beta)}$. Then the map $\overline{\beta}^{\vee}: \mathbb{Z}^{m}\rightarrow \overline{DG(\beta)}$ is given by $\overline{\beta}^{\vee}=(\overline{a}_1,\cdots,\overline{a}_m)$. For any column basis of the form $C=\{\overline{a}_{i_{1}},\cdots,\overline{a}_{i_{m-d}}\}$, there exist unique $\lambda_{1},\cdots,\lambda_{m-d}$ such that
$$a_{i_{1}}\lambda_{1}+\cdots+a_{i_{m-d}}\lambda_{m-d}=\overline{\theta}.$$
Let $\mathbb{C}[z_{1},\cdots,z_{m},w_{1},\cdots,w_{m}]$ be the coordinate ring of $\mathbb{C}^{2m}$. Let
$\sigma(C,\theta)=\{\overline{b}_{i_{j}}~|\lambda_{j}>0\}\sqcup\{\overline{b}'_{i_{j}}|~\lambda_{j}<0\},$
and
$C(\theta)=\{z_{i_{j}}~|\lambda_{j}>0\}\sqcup\{w_{i_{j}}|~\lambda_{j}<0\}$.
We set
\begin{equation}\label{irrelevant}
\mathbf{\mathcal{I}}_{\theta}:=<\prod
C(\theta)|~C~\text{is a column basis of}~\overline{\beta}^{\vee}>,
\end{equation}
and
\begin{equation}\label{fan}
\Sigma_{\theta}:=\{\overline{\sigma}(C,\theta):~C~\text{is a column basis of}~\overline{\beta}^{\vee}\},
\end{equation}
where $\overline{\sigma}(C,\theta)=
\{\overline{b}_{L,1},\cdots,\overline{b}_{L,m},\overline{b}'_{L,1},\cdots,\overline{b}'_{L,m}\}\setminus\sigma(C,\theta)$
is the complement of $\sigma(C,\theta)$ and corresponds to the maximal cones in $\Sigma_{\theta}$.
According to \cite{HS}, $\Sigma_{\theta}$ is the fan  of
Lawrence toric variety $X(\Sigma_{\theta})$ corresponding to
$\theta$ in the lattice $\overline{N}_{L}$.
The ideal
$\mathcal{I}_{\theta}$ is the irrelevant ideal of the fan $\Sigma_{\theta}$.
Then we have the Lawrence stacky fan $\mathbf{\Sigma_{\theta}}=(N_{L},\Sigma_{\theta},\beta_{L})$ introduced in \cite{JT}.

Applying $Hom_\mathbb{Z}(-,\mathbb{C}^{*})$ functor to (\ref{betaL}), we get
\begin{equation}\label{Lawrencemap}
\alpha_{h}: G\rightarrow (\mathbb{C}^{*})^{2m}.
\end{equation}
So $G$ acts on $\mathbb{C}^{2m}$ through $\alpha_{h}$. From Section 2,
$\mathcal{X}(\mathbf{\Sigma_{\theta}})=[(\mathbb{C}^{2m}\setminus V(\mathcal{I}_{\theta}))\slash G]$ whose coarse moduli space is the Lawrence toric variety $X(\Sigma_{\theta})=(\mathbb{C}^{2m}\setminus V(\mathcal{I}_{\theta}))\slash G$. Let $Y\subset \mathbb{C}^{2m}\setminus V(\mathcal{I}_{\theta})$
be the subvariety defined by the ideal:
\begin{equation}\label{ideal1}
I_{\beta^{\vee}}:=<\sum_{i=1}^{m}(\beta^{\vee})^{\star}(x)_{i}a_{ij}z_{i}w_{i}|\forall x\in DG(\beta)^{\star}>,
\end{equation}
where $(\beta^{\vee})^{\star}: DG(\beta)^{\star}\rightarrow \mathbb{Z}^{m}$ is the dual map of $\beta^{\vee}$ and $(\beta^{\vee})^{\star}(x)_{i}$ is the $i$-th component
of the vector $(\beta^{\vee})^{\star}(x)$.
From \cite{JT}, the hypertoric DM stack $\mathcal{M}(\mathcal{A})=[Y/G]$ whose coarse moduli space is the hypertoric variety $Y(\beta^{\vee},\theta)=Y\slash G$.

\begin{defn}(\cite{JT})
The Lawrence toric DM stack is  the toric DM stack $\mathcal{X}(\mathbf{\Sigma_{\theta}})$ corresponding to the Lawrence stacky fan $\mathbf{\Sigma_{\theta}}$.
\end{defn}

By \cite{HS},  $X(\Sigma_{\theta})$ is semi-projective. So the Lawrence toric DM stack $\mathcal{X}(\mathbf{\Sigma_{\theta}})$ is semi-projective by definition.

\subsection{Comparison of inertia stacks}
Next we compare the orbifold Chow ring of the hypertoric DM stack and the orbifold Chow ring of the Lawrence toric DM stack. First we compare the inertia stacks. From the map
$\beta: \mathbb{Z}^{m}\rightarrow N$ which is given by vectors $\{b_{1},\cdots,b_m\}$.  Let $Cone(\beta)$ be a partially ordered finite set of cones generated by $\overline{b}_{1},\cdots,\overline{b}_{m}$. The partial order is defined by: $\sigma\prec\tau$ if $\sigma$ is a face of $\tau$, and we have the minimum element $\hat{0}$ which is the cone consisting of the origin. Let $Cone(\overline{N})$ be the set of all convex polyhedral cones in the lattice $\overline{N}$. Then we have a map
$$C: Cone(\beta)\longrightarrow Cone(\overline{N}),$$ such that for any $\sigma\in Cone(\beta)$, $C(\sigma)$ is the cone in $\overline{N}$. Then $\Delta_{\mathbf{\beta}}:=(C,Cone(\beta))$ is a simplicial {\em multi-fan} in the sense of \cite{HM}.

For the multi-fan $\Delta_{\mathbf{\beta}}$, let  $Box(\Delta_{\mathbf{\beta}})$ be the set of pairs $(v,\sigma)$, where $\sigma$ is a cone in $\Delta_{\mathbf{\beta}}$, $v\in N$ such that $\overline{v}=\sum_{\rho_{i}\subseteq \sigma}\alpha_{i}b_{i}$ for $0<\alpha_{i}<1$. (Note that $\sigma$ is the minimal cone in $\Delta_{\mathbf{\beta}}$ satisfying the above condition.)  From \cite{JT}, an element $(v,\sigma)\in Box(\Delta_{\mathbf{\beta}})$ gives a component of the inertia stack $\mathcal{I}(\mathcal{M}(\mathcal{A}))$.
Also consider the set $Box(\mathbf{\Sigma_{\theta}})$ associated to the stacky fan $\mathbf{\Sigma_{\theta}}$, see Section 2.2 for its definition. An element $v\in Box(\mathbf{\Sigma_{\theta}})$ gives a component of the inertia stack $\mathcal{I}(\mathcal{X}(\mathbf{\Sigma_{\theta}}))$.

By the Lawrence lifting property, a vector $\overline{b}_{i}$ in $\overline{N}$ lifts to two vectors
$\overline{b}_{L,i},\overline{b}'_{L,i}$ in $\overline{N}_{L}$. Let $\{\overline{b}_{L,i_{1}},\cdots,\overline{b}_{L,i_{k}}, \overline{b}'_{L,i_{1}},\cdots,\overline{b}'_{L,i_{k}}\}$ be the Lawrence lifting of $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}\}$.

\begin{lem}\label{conemulti}
$\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}\}$ generate a cone  $\sigma$ in $\Delta_{\mathbf{\beta}}$ if and only if $\{\overline{b}_{L,i_{1}},\cdots,\overline{b}_{L,i_{k}}, \overline{b}'_{L,i_{1}},\cdots,\overline{b}'_{L,i_{k}}\}$  generate a cone $\sigma_{\theta}$ in $\Sigma_{\theta}$.
\end{lem}

\begin{pf}
Suppose $\sigma$ is a cone in $\Delta_{\mathbf{\beta}}$ generated by $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}\}$, it is contained in a top-dimensional cone $\tau$. Assume that $\tau$ is generated by $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}, \overline{b}_{i_{k+1}},\cdots,\overline{b}_{i_{d}}\}$. Let $C$ be the complement $\{\overline{b}_{1},\cdots,\overline{b}_{m}\}\setminus \tau$. Then $C$ corresponds to a column basis of  $\overline{\beta}^{\vee}$ in the map $\overline{\beta}^{\vee}: \mathbb{Z}^{m}\rightarrow \overline{DG(\beta)}$. By the definition of $\Sigma_{\theta}$ in (\ref{fan}), $C$ corresponds to a maximal cone $\tau_{\theta}$ in $\Sigma_{\theta}$ which contains
the rays generated by $\{\overline{b}_{L,i_{1}},\cdots,\overline{b}_{L,i_{k}},
\overline{b}'_{L,i_{1}},\cdots,\overline{b}'_{L,i_{k}}\}$. Thus these rays generate a cone $\sigma_{\theta}$
in $\Sigma_{\theta}$.

Conversely, suppose $\sigma_{\theta}$ is a cone in $\Sigma_{\theta}$ generated by $\{\overline{b}_{L,i_{1}},\cdots,\overline{b}_{L,i_{k}}, \overline{b}'_{L,i_{1}},\cdots,\overline{b}'_{L,i_{k}}\}$. Using the similar method above we prove that $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}\}$ must be contained in a top-dimensional cone of $\Delta_{\mathbf{\beta}}$. So $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}}\}$ generate a cone $\sigma$ in $\Delta_{\mathbf{\beta}}$.
\end{pf}

\begin{lem}\label{box}
There is an one-to-one correspondence between the elements in $Box(\mathbf{\Sigma_{\theta}})$ and the elements in $Box(\Delta_{\mathbf{\beta}})$. Moreover, their degree shifting numbers coincide.
\end{lem}

\begin{pf}
First the torsion elements in $Box(\mathbf{\Sigma_{\theta}})$ and $Box(\Delta_{\mathbf{\beta}})$ are both
isomorphic to $\mu=ker(\alpha)=ker(\alpha_{h})$ in (\ref{exact3}) and (\ref{Lawrencemap}).
Let $(v,\sigma)\in Box(\Delta_{\mathbf{\beta}})$ with $\overline{v}=\sum_{\rho_i\subseteq \sigma}\alpha_{i}\overline{b}_{i}$.
Then $v$ may be identified with an element (which we ambiguously denote by) $v\in G:=Hom_\mathbb{Z}(DG(\beta),\mathbb{C}^*)$. Certainly $v$ fixes a point in $\mathbb{C}^{m}$. Consider the map $\alpha$ in (\ref{exact3}), put $\alpha(v)=(\alpha^{1}(v),\cdots,\alpha^{m}(v))$. Then $\alpha^{i}(v)\neq 1$ if $\rho_i\subseteq\sigma$, and $\alpha^{i}(v)= 1$ otherwise. By Lemma \ref{conemulti}, let $\{\overline{b}_{L,i},\overline{b}'_{L,i}:i=1,\cdots,|\sigma|\}$ be the Lawrence lifting of $\{\overline{b}_{i}\}_{\rho_i\subseteq\sigma}$. Since the action of $v$ on $\mathbb{C}^{2m}$ is given by $(v,v^{-1})$, $v$ fixes a point in $\mathbb{C}^{2m}$ and yields an element $v_\theta$ in $Box(\mathbf{\Sigma_\theta})$.
From the map (\ref{Lawrencemap}), let
\begin{equation}\label{vtheta}
\alpha_{h}(v_{\theta})=(\alpha^{1}_{h}(v_{\theta}),\cdots,\alpha^{m}_{h}(v_{\theta}),
\alpha^{m+1}_{h}(v_{\theta}),\cdots,\alpha^{2m}_{h}(v_{\theta})).
\end{equation}
Then $\alpha_{h}^{i}(v_{\theta})\neq 1$ and $\alpha_{h}^{i+m}(v_{\theta})\neq 1$ if $\rho_i\subseteq\sigma$; $\alpha_{h}^{i}(v_{\theta})= \alpha_{h}^{i+m}(v_{\theta})= 1$ otherwise.  So $\sigma_{\theta}(\overline{v}_{\theta})=\{\overline{b}_{L,i},\overline{b}'_{L,i}:i=1,\cdots,|\sigma|\}$ is the minimal cone in $\Sigma_{\theta}$ containing $\overline{v}_{\theta}$. Furthermore, $\overline{v}_{\theta}=\sum_{\rho_i\subseteq \sigma}\alpha_{i}\overline{b}_{L,i}+ \sum_{\rho_i\subseteq \sigma}(1-\alpha_{i})\overline{b}'_{L,i}$.

Conservely, given an element $v_{\theta}\in Box(\mathbf{\Sigma_{\theta}})$, let $\sigma_{\theta}(\overline{v}_{\theta})$ be the minimal cone in $\Sigma_{\theta}$ containing $\overline{v}_{\theta}$.
Then from the action of $G$ on $\mathbb{C}^{2m}$ and (\ref{vtheta}), we have $\alpha^{i}_{h}(v_{\theta})=(\alpha^{i+m}_{h}(v_{\theta}))^{-1}$. If $\alpha^{i}_{h}(v_{\theta})\neq 1$, then $\alpha^{i+m}_{h}(v_{\theta})\neq 1$, which means that  $\overline{b}_{L,i}, \overline{b}_{L,i+m}\in \sigma_{\theta}(\overline{v}_{\theta})$. The cone $\sigma_{\theta}(\overline{v}_{\theta})$ is the one in $\Sigma_{\theta}$ containing $\overline{b}_{L,i}, \overline{b}_{L,i+m}$'s satisfying this condition.
Then $\overline{v}_{\theta}=\sum_{i}(\alpha_{i}\overline{b}_{L,i}+(1-\alpha_{i})\overline{b}^{'}_{L,i})$.
By Lemma \ref{conemulti},
$\sigma_{\theta}(\overline{v}_{\theta})$ is the Lawrence lifting of a cone $\sigma$ generated by the $\{\overline{b}_{i}\}$'s in $\Delta_{\mathbf{\beta}}$. Let $v=\sum_{\rho_i\subseteq\sigma}\alpha_{i}b_{i}$. So it also determines an element $(v,\sigma)\in Box(\Delta_{\mathbf{\beta}})$.
\end{pf}

For $(v_{1},\sigma_{1}),(v_{2},\sigma_{2}),(v_{3},\sigma_{3})\in Box(\Delta_{\mathbf{\beta}})$, let $\sigma(\overline{v}_{1}, \overline{v}_{2},\overline{v}_{3})$ be the miniaml cone containing $\overline{v}_{1}, \overline{v}_{2},\overline{v}_{3}$ in $\Delta_{\mathbf{\beta}}$ such that $\overline{v}_{1}+\overline{v}_{2}+\overline{v}_{3}=\sum_{\rho_i\subseteq \sigma(\overline{v}_{1}, \overline{v}_{2},\overline{v}_{3})}a_{i}\overline{b}_{i}$ and $a_{i}=1,2$. Let $v_{\theta,1}, v_{\theta,2},v_{\theta,3}$ be the corresponding elements in  $Box(\mathbf{\Sigma_{\theta}})$ and $\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3})$ the minimal cone containing $\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3}$ in $\mathbf{\Sigma_{\theta}}$. Then by Lemmas \ref{conemulti} and \ref{box},
$\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3})$ is the Lawrence lifting of  $\sigma(\overline{v}_{1}, \overline{v}_{2},\overline{v}_{3})$. Suppose that $\sigma$
is generated by $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{s}}\}$, then  $\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3})$ is generated by $\{\overline{b}_{L,i_{1}},\cdots,\overline{b}_{L,i_{s}},\overline{b}^{'}_{L,i_{1}},\cdots,\overline{b}^{'}_{L,i_{s}}\}$, the Lawrence lifting
of $\{\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{s}}\}$. Let $\{\overline{b}_{j_{1}},\cdots,\overline{b}_{j_{m-l-s}}\}$ be the rays
not in $\sigma\cup link(\sigma)$, we have the Lawrence lifting
$\{\overline{b}_{L,j_{1}},\cdots,\overline{b}_{L,j_{m-l-s}},\overline{b}^{'}_{L,j_{1}},\cdots,\overline{b}^{'}_{L,j_{m-l-s}}\}$. Then
from the definition of Lawrence fan $\Sigma_{\theta}$ in (\ref{fan}), we have the following lemma:
\begin{lem}\label{keycone}
There exist $m-l-s$  vectors in $\{\overline{b}_{L,j_{1}},\cdots,\overline{b}_{L,j_{m-l-s}},\overline{b}^{'}_{L,j_{1}},\cdots,\overline{b}^{'}_{L,j_{m-l-s}}\}$
such that the rays they generate plus the rays in $\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3})$
generate a cone $\sigma_{\theta}$ in $\Sigma_{\theta}$. $\square$
\end{lem}

\begin{prop}\label{3-twisted-sector}
The stack $\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})$ is also a Lawrence toric DM stack.
\end{prop}

\begin{pf}
For simplicity, put $\sigma:=\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})$. Suppose there are
$l$ rays in the $link(\sigma)$. Then by Lemma \ref{conemulti} there
are $2l$ rays in $link(\sigma_{\theta})$, the Lawrence lifting of $link(\sigma)$.
Let $s:=|\sigma|$, then $2s+m-l-s=|\sigma_{\theta}|$. Applying Gale dual to the diagrams
\[
\begin{CD}
0 @ >>>\mathbb{Z}^{s}@ >>> \mathbb{Z}^{l+s}@ >>> \mathbb{Z}^{l} @
>>> 0\\
&& @VV{\beta_{\sigma}}V@VV{\widetilde{\beta}}V@VV{\beta(\sigma)}V \\
0@ >>>N_{\sigma} @ >{}>>N@ >>> N(\sigma) @>>> 0,
\end{CD}
\]
and
\[
\begin{CD}
0 @ >>>\mathbb{Z}^{l+s}@ >>> \mathbb{Z}^{m}@ >>> \mathbb{Z}^{m-l-s} @
>>> 0\\
&& @VV{\widetilde{\beta}}V@VV{\beta}V@VV{}V \\
0@ >>>N @ >{\cong}>>N@ >>> 0 @>>> 0
\end{CD}
\]
yields
\begin{equation}\label{3-sector2}
\begin{CD}
0 @ >>>\mathbb{Z}^{l}@ >>> \mathbb{Z}^{l+s}@ >>> \mathbb{Z}^{s} @
>>> 0\\
&& @VV{\beta(\sigma)^{\vee}}V@VV{\widetilde{\beta}^{\vee}}V@VV{\beta_{\sigma}^{\vee}}V \\
0@ >>>DG(\beta(\sigma)) @ >{\varphi_{1}}>>DG(\widetilde{\beta})@ >>> DG(\beta_{\sigma})
@>>> 0,
\end{CD}
\end{equation}
and
\begin{equation}\label{3-sector22}
\begin{CD}
0 @ >>>\mathbb{Z}^{m-l-s}@ >>> \mathbb{Z}^{m}@ >>> \mathbb{Z}^{l+s} @
>>> 0\\
&& @VV{\cong}V@VV{\beta^{\vee}}V@VV{\widetilde{\beta}^{\vee}}V \\
0@ >>>\mathbb{Z}^{m-l-s} @ >{}>>DG(\beta)@ >{\varphi_{2}}>> DG(\widetilde{\beta})
@>>> 0.
\end{CD}
\end{equation}
Since $\mathbb{Z}^{s}\cong N_{\sigma}$, we have that $DG(\beta_{\sigma})=0$.
We add two exact sequences
$$0\longrightarrow \mathbb{Z}^{l}\longrightarrow\mathbb{Z}^{m}\longrightarrow\mathbb{Z}^{m-l}\longrightarrow 0,$$
and
$$0\longrightarrow 0\longrightarrow\mathbb{Z}^{m}\longrightarrow\mathbb{Z}^{m}\longrightarrow 0,$$
on the rows of the diagrams (\ref{3-sector2}),(\ref{3-sector22}) and make suitable maps to  the
Gale duals we get
\begin{equation}\label{3-sectors}
\begin{CD}
0 @ >>>\mathbb{Z}^{2l}@ >>> \mathbb{Z}^{l+s+m}@ >>>
\mathbb{Z}^{s+m-l} @
>>> 0\\
&& @VV{(\beta(\sigma)^{\vee},-\beta(\sigma)^{\vee})}V@VV{(\widetilde{\beta}^{\vee},-\beta^{\vee})}V@VV{0}V \\
0@ >>>DG(\beta(\sigma)) @ >{\cong}>>DG(\widetilde{\beta})@ >>> 0
@>>> 0,
\end{CD}
\end{equation}
and
\begin{equation}\label{3-sectors2}
\begin{CD}
0 @ >>>\mathbb{Z}^{m-l-s}@ >>> \mathbb{Z}^{2m}@ >>>
\mathbb{Z}^{l+s+m} @
>>> 0\\
&& @VV{\cong}V@VV{(\beta^{\vee},-\beta^{\vee})}V@VV{(\widetilde{\beta}^{\vee},-\beta^{\vee})}V \\
0@ >>>\mathbb{Z}^{m-l-s} @ >{}>>DG(\beta)@ >>> DG(\widetilde{\beta})
@>>> 0.
\end{CD}
\end{equation}
Applying Gale dual to (\ref{3-sectors}), (\ref{3-sectors2}) we get
\[
\begin{CD}
0 @ >>>\mathbb{Z}^{s+m-l}@ >>> \mathbb{Z}^{l+s+m}@ >>> \mathbb{Z}^{2l} @ >>> 0\\
&& @VV{\cong}V@VV{\widetilde{\beta}_{L}}V@VV{\beta_{L}(\sigma_{\theta})}V \\
0@ >>>\mathbb{Z}^{s+m-l} @ >{}>>\widetilde{N}_{L}@ >>> N_{L}(\sigma_{\theta})
@>>> 0,
\end{CD}
\]
 and
\[
\begin{CD}
0 @ >>>\mathbb{Z}^{l+s+m}@ >>> \mathbb{Z}^{2m}@ >>> \mathbb{Z}^{m-l-s} @ >>> 0\\
&& @VV{\widetilde{\beta}_{L}}V@VV{\beta_{L}}V@VV{0}V \\
0@ >>>\widetilde{N}_{L} @ >{\cong}>>N_{L}@ >>> 0
@>>> 0.
\end{CD}
\]
For the generic element
$\theta$, from them map $\varphi_{2}$ in (\ref{3-sector22}),
$\theta$ induces $\widetilde{\theta}\in DG(\widetilde{\beta})$, and from the isomorphism
$\varphi_{1}$ in (\ref{3-sector2}),  $\widetilde{\theta}=\theta(\sigma)\in DG(\beta(\sigma))$.
So we a quotient stacky hyperplane arrangement $\mathcal{A}(\sigma)=(N(\sigma),\beta(\sigma),\theta(\sigma))$.
From the above diagrams  we see that the quotient fan
$\Sigma_{\theta}/\sigma_{\theta}$ in
$\overline{N}_{L}(\sigma_{\theta})$ also comes from a Lawrence
construction of the map $\beta(\sigma)^{\vee}:
\mathbb{Z}^{l}\rightarrow DG(\beta(\sigma))$.  Let
$X(\sigma)=\mathbb{C}^{2l}\setminus
V(\mathcal{I}_{\theta(\sigma)})$, where
$\mathcal{I}_{\theta(\sigma)}$ is the irrelevant ideal  of the
quotient fan $\Sigma_{\theta}\slash \sigma_{\theta}$. Let
$G(\sigma)=Hom_\mathbb{Z}(DG(\beta(\sigma)),\mathbb{C}^{*})$. The
stack
$\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})=[X(\sigma)/G(\sigma)]$
is a Lawrence toric Deligne-Mumford stack.
\end{pf}

\begin{cor}\label{sectors}
$\mathcal{M}(\mathcal{A}(\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})))$ is the hypertoric DM stack associated to the quotient Lawrence toric DM stack
$\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})$.
\end{cor}

\begin{pf}
$\mathcal{M}(\mathcal{A}(\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})))$ is constructed in \cite{JT} as a quotient stack
$[Y(\sigma)/G(\sigma)]$, where $Y(\sigma)\subset X(\sigma)$ is defined by $I_{\beta(\sigma)^{\vee}}$, which is
the ideal in (\ref{ideal1}) corresponding to the map $\beta(\sigma)^{\vee}$ in (\ref{3-sector2}). So the
stack $\mathcal{M}(\mathcal{A}(\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})))$ is the associated hypertoric DM stack in
the Lawrence toric DM stack $\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})$.
\end{pf}

\begin{rmk}
For any $v_{\theta}\in Box(\mathbf{\Sigma_{\theta}})$, let $v^{-1}_{\theta}$ be its inverse. We have
the quotient Lawrence toric stack $\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})$.
Let $(v,\sigma)$ be the corresponding element in $Box(\Delta_{\mathbf{\beta}})$, then
$$\mathcal{M}(\mathcal{A}(\sigma(\overline{v},
\overline{v}^{-1},1)))\cong \mathcal{M}(\mathcal{A}(\sigma)).$$
By Proposition \ref{3-twisted-sector} and Corollary \ref{sectors}, the twisted sector
$\mathcal{M}(\mathcal{A}(\sigma))$ is the associated hypertoric DM stack
of the Lawrence toric DM stack $\mathcal{X}(\mathbf{\Sigma_{\theta}}/\sigma_{\theta})$.
\end{rmk}

\begin{rmk}
From Lemma \ref{keycone}, the cone $\sigma_{\theta}$ is not the minimal cone
$\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3})$ containing
$\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3}$ in $\Sigma_{\theta}$.
So $\mathcal{X}(\Sigma_{\theta}/\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3}))$
is not a Lawrence toric DM stack. But from the construction of Lawrence toric DM stack,
the quotient stack $\mathcal{X}(\Sigma_{\theta}/\sigma(\overline{v}_{\theta,1}, \overline{v}_{\theta,2},\overline{v}_{\theta,3}))$
is homotopy equivalent to the quotient stack $\mathcal{X}(\Sigma_{\theta}/\sigma_{\theta})$.
Since we do not need this to compare the orbifold Chow ring, we omit the details.
\end{rmk}

\subsection{Comparison of orbifold Chow rings}
Recall that $N_{L}=\overline{N}_{L}\oplus N_{L,tor}$, where $N_{L,tor}$ is the torsion subgroup
of $N_{L}$.  Let $N_{\Sigma_{\theta}}=N_{L,tor}\oplus |\Sigma_{\theta}|$.  By Theorem 1.1, we have

\begin{prop}\label{orbifoldlawrence}
The orbifold Chow ring $A^{*}_{orb}(\mathcal{X}(\mathbf{\Sigma}_\theta))$ of the
Lawrence toric  DM stack $\mathcal{X}(\mathbf{\Sigma_{\theta}})$ is isomorphic to the ring
\begin{equation}\label{chowringlawrence}
\frac{\mathbb{Q}[N_{\Sigma_{\theta}}]}
{\{\sum_{i=1}^{m}e(b_{L,i})y^{b_{L,i}}+\sum_{i=1}^{m}e(b'_{L,i})y^{b'_{L,i}}:e\in N_{L}^{\star}\}}.
\end{equation}~ $\square$
\end{prop}

Recall in \cite{JT} that  for any $c\in N$, there is a cone
$\sigma\in \Delta_\mathbf{\beta}$ such that
$\overline{c}=\sum_{\rho_{i}\subseteq \sigma}\alpha_{i}\overline{b}_{i}$ where
$\alpha_{i}>0$ are  rational numbers. Let $N^{\Delta_\mathbf{\beta}}$ denote
all the pairs $(c,\sigma)$. Then $N^{\Delta_\mathbf{\beta}}$ gives rise a
group ring
$$\mathbb{Q}[\Delta_\mathbf{\beta}]=\bigoplus_{(c,\sigma)\in N^{\Delta_\mathbf{\beta}}}\mathbb{Q}\cdot y^{(c,\sigma)},$$
where $y$ is a formal variable.
For any $(c,\sigma)\in N^{\Delta_\mathbf{\beta}}$, there exists a unique element
$(v,\tau)\in Box(\Delta_\mathbf{\beta})$ such that $\tau\subseteq\sigma$ and
$c=v+\sum_{\rho_{i}\subseteq \sigma}m_{i}b_{i}$,
where
$m_{i}$ are nonnegative integers.  We call $(v,\tau)$ the fractional part of $(v,\sigma)$. We define the $ceiling ~function$ for fans.
For $(c,\sigma)$ define the ceiling function $\lceil c \rceil_{\sigma}$ by
$\lceil c \rceil_{\sigma}=\sum_{\rho_{i}\subseteq \tau}b_{i}+\sum_{\rho_{i}\subseteq \sigma}m_{i}b_{i}$. Note that
if $\overline{v}=0$, $\lceil c \rceil_{\sigma}=\sum_{\rho_{i}\subseteq \sigma}m_{i}b_{i}$.
For two pairs  $(c_1,\sigma_1)$, $(c_2,\sigma_2)$, if $\sigma_1\cup\sigma_2$ is a cone in $\Delta_\mathbf{\beta}$, define
$\epsilon(c_1,c_2):=\lceil c_1 \rceil_{\sigma_{1}}+\lceil c_2 \rceil_{\sigma_{2}}-\lceil c_1+c_2 \rceil_{\sigma_{1}\cup\sigma_2}$.
Let $\sigma_{\epsilon}\subseteq\sigma_1\cup\sigma_2$ be the minimal cone in $\Delta_\mathbf{\beta}$ containing $\epsilon(c_1,c_2)$ so that
$(\epsilon(c_1,c_2),\sigma_{\epsilon})\in N^{\Delta_\mathbf{\beta}}$. We define the grading on $\mathbb{Q}[\Delta_{\mathbf{\beta}}]$ as follows.
For any  $(c,\sigma)$, write $c=v+\sum_{\rho_{i}\subseteq \sigma}m_{i}b_{i}$,  then
$deg(y^{(c,\sigma)})=|\tau|+\sum_{\rho_{i}\subseteq\sigma}m_{i}$, where
$|\tau|$ is the dimension of $\tau$.
By abuse of notation, we write $y^{(b_{i},\rho_i)}$ as $y^{b_{i}}$.
The multiplication
is defined by
\begin{equation}\label{product1}
y^{(c_{1},\sigma_{1})}\cdot y^{(c_{2},\sigma_{2})}:=
\begin{cases}
(-1)^{|\sigma_{\epsilon}|}y^{(c_{1}+c_{2}+\epsilon(c_1,c_2),\sigma_1\cup\sigma_2)}&\text{if
$\sigma_{1}\cup\sigma_{2}$ is a cone in $\Delta_{\mathbf{\beta}}$}\,,\\
0&\text{otherwise}\,.
\end{cases}
\end{equation}
From the property of $ceiling~ function$ we check that the multiplication is commutative and associative. So $\mathbb{Q}[\Delta_\mathbf{\beta}]$ is
a unital associative commutative	ring.
In \cite{JT}, it is shown that
\begin{equation}\label{chowringhyper}
A^{*}_{orb}(\mathcal{M}(\mathcal{A}))\cong \frac{\mathbb{Q}[\Delta_{\mathbf{\beta}}]}{\{\sum_{i=1}^{m}e(b_{i})y^{b_{i}}: e\in N^{\star}\}}.
\end{equation}

Consider the map $\beta: \mathbb{Z}^{m}\rightarrow N$ which is given by the vectors
$\{b_{1},\cdots,b_{m}\}$.
We take $\{1,\cdots,m\}$ as the vertex set of the {\em matroid complex}
$M_{\beta}$, defined from $\beta$ by requiring that $F\in
M_{\beta}$ iff the  vectors
$\{\overline{b}_{i}\}_{i\in F}$ are linearly independent in $\overline{N}$.
A face $F\in M_{\beta}$ corresponds
to a cone in $\Delta_{\mathbf{\beta}}$ generated by $\{\overline{b}_{i}\}_{i\in F}$.  By
\cite{S}, the ``Stanley-Reisner'' ring of the matroid
$M_{\beta}$ is
$$\mathbb{Q}[M_{\beta}]=\frac{\mathbb{Q}[y^{b_{1}},\cdots,y^{b_{m}}]}{I_{M_{\beta}}},$$
where $I_{M_{\beta}}$ is the matroid ideal generated by the set
of  square-free monomials
$$\{y^{b_{i_{1}}}\cdots y^{b_{i_{k}}}|
\overline{b}_{i_{1}},\cdots,\overline{b}_{i_{k}} ~\text{linearly
dependent in}~\overline{N}\}.$$
It is proved in  \cite{JT} that ,
$$\mathbb{Q}[\Delta_{\mathbf{\beta}}]\cong\bigoplus_{(v,\sigma)\in Box(\Delta_{\mathbf{\beta}})}y^{(v,\sigma)}\cdot \mathbb{Q}[M_{\beta}].$$
For any  $(v_{1},\sigma_{1}),(v_{2},\sigma_{2})\in
Box(\Delta_{\mathbf{\beta}})$,
let $(v_{3},\sigma_{3})$ be the unique element in $Box(\Delta_{\mathbf{\beta}})$
such that $v_1+v_2+v_3\equiv 0$ in the local group given by $\sigma_1\cup\sigma_2$, where
$\equiv 0$ means that there exists a cone  $\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})$ in
$\Delta_{\mathbf{\beta}}$ such that
$\overline{v}_{1}+\overline{v}_{2}+\overline{v}_{3}=\sum_{\rho_{i}\subseteq\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})}a_{i}\overline{b}_{i}$,
where  $a_{i}=1 ~\text{or}~ 2$.  Let $\overline{v}_1=\sum_{\rho_j\subseteq\sigma_1}\alpha_{j}^{1}\overline{b}_{j}$,
$\overline{v}_2=\sum_{\rho_j\subseteq\sigma_2}\alpha_{j}^{2}\overline{b}_{j}$,
$\overline{v}_3=\sum_{\rho_j\subseteq\sigma_3}\alpha_{j}^{3}\overline{b}_{j}$ with
$0<\alpha_{j}^{1},\alpha_{j}^{2},\alpha_{j}^{3}<1$. Let $I$ be the set of $i$ such that
$a_{i}=1$ and $\alpha_{j}^{1},\alpha_{j}^{2},\alpha_{j}^{3}$ exist, $J$  the set of $j$ such that $\rho_{j}$ belongs
to $\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})$ but not $\sigma_{3}$.
If $(v,\sigma)\in Box(\Delta_\mathbf{\beta})$, let
$(\check{v},\sigma)$ be the inverse of $(v,\sigma)$. Except torsion elements, equivalently, if $\overline{v}=\sum_{\rho_{i}\subseteq \sigma}\alpha_{i}\overline{b}_{i}$ for
$0<\alpha_{i}<1$, then $\check{\overline{v}}=\sum_{\rho_{i}\subseteq \sigma}(1-\alpha_{i})\overline{b}_{i}$.
By abuse of notation, we write $y^{(b_{i},\rho_i)}$ as $y^{b_{i}}$.
We have that  $v_{1}+v_{2}=\check{v}_{3}+\sum_{a_{i}=2}b_{i}+\sum_{j\in J}b_{j}$. From (\ref{product1}), Lemma 5.11
and Lemma 5.12 in \cite{JT},
if $\overline{v}_{1},\overline{v}_{2}\neq 0$, we have
$$
\lceil v_1\rceil_{\sigma_1}+\lceil v_2\rceil_{\sigma_2}-\lceil v_1+v_2\rceil_{\sigma_1\cup\sigma_2}=
\begin{cases}
\sum_{i\in I}b_{i}+\sum_{j\in J}b_{j}&\text{if
$\overline{v}_{1}\neq\check{\overline{v}}_{2}$}\,,\\
\sum_{j\in J}b_{j}&\text{if
$\overline{v}_{1}=\check{\overline{v}}_{2}$}\,.\\
\end{cases}
$$
So it is easy to check that the multiplication
$y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}$ can be written as
\begin{equation}\label{product}
\begin{cases}
(-1)^{|I|+|J|}y^{(\check{v}_{3},\sigma_{3})}\cdot\prod_{a_{i}=2}
y^{b_{i}}\cdot\prod_{i\in I}
y^{b_{i}}\cdot \prod_{j\in J}y^{2b_{j}}&\text{if
$\overline{v}_{1},\overline{v}_{2}\in\sigma$ for $\sigma \in\Delta_{\mathbf{\beta}}$ and $\overline{v}_{1}\neq \check{\overline{v}}_{2}$}\,,\\
(-1)^{|J|} \prod_{j\in J}y^{2b_{j}}&\text{if
$\overline{v}_{1},\overline{v}_{2}\in\sigma$ for $\sigma \in\Delta_{\mathbf{\beta}}$ and $\overline{v}_{1}=\check{\overline{v}}_{2}$}\,,\\
0&\text{otherwise}\,.
\end{cases}
\end{equation}

The following is the main result of this Section.

\begin{thm}\label{main2}
There is an isomorphism of orbifold Chow rings $A_{orb}^{*}(\mathcal{X}(\mathbf{\Sigma_{\theta}}))\cong A_{orb}^{*}(\mathcal{M}(\mathcal{A}))$.
\end{thm}

\begin{pf}
The ring $\mathbb{Q}[N_{\Sigma_{\theta}}]$ is generated by $\{y^{b_{L,i}},y^{b'_{L,i}}: i=1,\cdots,m\}$
and $y^{v_{\theta}}$ for $v_{\theta}\in Box(\mathbf{\Sigma_{\theta}})$ by the definition. By Lemma \ref{box}, define a morphism
$$\phi: \mathbb{Q}[N_{\Sigma_{\theta}}]\rightarrow\mathbb{Q}[\Delta_{\mathbf{\beta}}] $$
by $y^{b_{L,i}}\mapsto y^{b_{i}}, y^{b'_{L,i}}\mapsto -y^{b_{i}}$ and $y^{v_{\theta}}\mapsto y^{(v,\sigma)}$. By \cite{HS}, the ideal $\mathcal{I}_{\theta}$ goes to the ideal $I_{M_{\beta}}$ and the relation
$\{\sum_{i=1}^{m}e(b_{L,i})y^{b_{L,i}}+\sum_{i=1}^{m}e(b'_{L,i})y^{b'_{L,i}}:e\in N_{L}^{\star}\}$
goes to the relation $\{\sum_{i=1}^{m}e(b_{i})y^{b_{i}}:e\in N^{\star}\}$. Thus the two rings are isomorphic
as modules.

It remains to check the multiplications. For any $y^{v_{\theta}}$ and $y^{b_{L,i}}$ or $y^{b'_{L,i}}$, let
$y^{(v,\sigma)}$ be the corresponding element in $\mathbb{Q}[\Delta_{\mathbf{\beta}}]$.
By the property of $v_{\theta}$ and Lemma \ref{box}, the minimal cone in
$\Sigma_{\theta}$ containing $\overline{v}_{\theta}, \overline{b}_{L,i}$ must contains $\overline{b}'_{L,i}$.
By Lemma \ref{conemulti}, there is a cone in $\Delta_{\mathbf{\beta}}$ containing $\overline{v}, \overline{b}_{i}$.
In this way, $y^{v_{\theta}}\cdot y^{b_{L,i}}$ goes to $y^{(v,\sigma)}\cdot y^{b_{i}}$ and
$y^{v_{\theta}}\cdot y^{b'_{L,i}}$ goes to $-y^{(v,\sigma)}\cdot y^{b_{i}}$.
If there is no cone in $\Sigma_{\theta}$
containing $\overline{v}_{\theta}, \overline{b}_{L,i},\overline{b}'_{L,i}$, then by Lemma \ref{conemulti} there is no cone in $\Delta_{\mathbf{\beta}}$ containing $\overline{v}, \overline{b}_{i}$. So
$y^{v_{\theta}}\cdot y^{b_{L,i}}=0$ goes to $y^{(v,\sigma)}\cdot y^{b_{i}}=0$ and
$y^{v_{\theta}}\cdot y^{b'_{L,i}}=0$ goes to $-y^{(v,\sigma)}\cdot y^{b_{i}}=0$.

For any $y^{v_{\theta,1}},y^{v_{\theta,2}}$, let $y^{(v_{1},\sigma_{1})},y^{(v_{2},\sigma_{2})}$ be the corresponding
elements in $\mathbb{Q}[\Delta_{\mathbf{\beta}}]$. If there is no cone
in $\Sigma_{\theta}$ containing $\overline{v}_{\theta,1},\overline{v}_{\theta,2}$, then by Lemmas \ref{conemulti}
and \ref{box}, there is no cone in $\Delta_{\mathbf{\beta}}$ containing $\overline{v}_{1},\overline{v}_{2}$. So
$y^{v_{\theta,1}}\cdot y^{v_{\theta,2}}=0$ goes to $y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}=0$.
Suppose there
is a cone  containing $\overline{v}_{\theta,1},\overline{v}_{\theta,2}$, let
$v_{\theta,3}\in Box(\mathbf{\Sigma}_{\theta})$ such that $v_{\theta,1}+v_{\theta,2}+v_{\theta,3}\equiv 0$.
Let $\sigma(\overline{v}_{\theta,1},
\overline{v}_{\theta,2},\overline{v}_{\theta,3})$ be the minimal cone containing $\overline{v}_{\theta,1},
\overline{v}_{\theta,2},\overline{v}_{\theta,3}$ in $\Sigma_{\theta}$. Then by Lemmas \ref{conemulti}
and \ref{box},
$\sigma(\overline{v}_{\theta,1},
\overline{v}_{\theta,2},\overline{v}_{\theta,3})$ is the Lawrence lifting of $\sigma(\overline{v}_{1},
\overline{v}_{2},\overline{v}_{3})$ for  $(v_{1},\sigma_{1}),(v_{2},\sigma_{2}),(v_{3},\sigma_{3})\in Box(\Delta_{\mathbf{\beta}})$.
So we may write
$\overline{v}_{\theta,1}+\overline{v}_{\theta,2}+\overline{v}_{\theta,3}=\sum_{\rho_{i}\subseteq \sigma(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})}a_{i}\overline{b}_{L,i}
+\sum_{\rho_{i}\subseteq \sigma(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})}a'_{i}\overline{b}'_{L,i}$.
The corresponding $\overline{v}_{1}+\overline{v}_{2}+\overline{v}_{3}=
\sum_{\rho_{i}\subseteq \sigma(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})}a_{i}\overline{b}_{i}$.
Let
$(\check{v},\sigma)$ be  the inverse of $(v,\sigma)$ in
$Box(\Delta_{\mathbf{\beta}})$, i.e. if $v$ is nontorsion and
$\overline{v}=\sum_{\rho_{i}\subseteq \sigma}\alpha_{i}\overline{b}_{i}$
for $0<\alpha_{i}<1$, then  $\check{\overline{v}}=\sum_{\rho_{i}\subseteq
\sigma}(1-\alpha_{i})\overline{b}_{i}$. The $\check{v}_{\theta}$ is defined similarly
in $Box(\mathbf{\Sigma_{\theta}})$. The notation  $J$ represents the set of $j$ such that $\rho_{j}$ belongs
to $\sigma(\overline{v}_{1},\overline{v}_{2},\overline{v}_{3})$ but not $\sigma_{3}$,   the
corresponding $\rho_{L,j},\rho'_{L,j}$ belong
to $\sigma(\overline{v}_{\theta,1},\overline{v}_{\theta,2},\overline{v}_{\theta,3})$ but not $\sigma(\overline{v}_{\theta,3})$.

If some $\overline{v}_{\theta,i}=0$ which means that $v_{\theta,i}$ is a torsion. Then
from Lemma (\ref{box}) the corresponding
$v$ is also a torsion element. In this case we know that the orbifold cup product
$y^{v_{\theta,1}}\cdot y^{v_{\theta,2}}$ is the usual product, and under the
map $\phi$,  is equal to
$y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}$.

If $\overline{v}_{\theta,1}=\check{\overline{v}}_{\theta,2}$, then $\overline{v}_{\theta,3}=0$ and the obstruction bundle over the corresponding 3-twisted sector is zero.
The set $J$ is the set $j$ such that $\rho_j$ belongs to $\sigma(\overline{v}_{\theta,1})$.
So from \cite{BCS}, we have
$$y^{v_{\theta,1}}\cdot y^{v_{\theta,2}}=
\prod_{j\in J}y^{b_{L,j}}\cdot y^{b'_{L,j}}.$$
Under the map $\phi$ we see that $y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}$ is equal to the second line in the  product
(\ref{product}).

If $\overline{v}_{\theta,1}\neq\check{\overline{v}}_{\theta,2}$, then $\overline{v}_{\theta,3}\neq 0$ and the obstruction bundle
is given by Proposition \ref{obstructionbdle}.  If all  $\alpha_{j}^{1},\alpha_{j}^{2},\alpha_{j}^{3}$ exist, the coefficients $a_{i}$ and $a'_{i}$ satisfy that if $a_{i}=1$ then
$a'_{i}=2$, and if $a_{i}=2$ then
$a'_{i}=1$. So
from \cite{BCS},
$$y^{v_{\theta,1}}\cdot y^{v_{\theta,2}}=
y^{\check{v}_{\theta,3}}\cdot\prod_{a_{i}=2}
y^{b_{L,i}}\cdot\prod_{i\in I}
y^{b'_{L,i}}\cdot \prod_{j\in J}y^{b_{L,j}}\cdot y^{b'_{L,j}}.$$
Under the map $\phi$ we see that $y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}$ is equal to the first line in the  product
(\ref{product}). By Lemma \ref{box}, the box elements have the same orbifold degrees.
By Corollary \ref{sectors} and the definition of orbifold cup product in (\ref{cupproduct}),
the products $y^{v_{\theta,1}}\cdot y^{v_{\theta,2}}$ and $y^{(v_{1},\sigma_{1})}\cdot y^{(v_{2},\sigma_{2})}$ have
the same degrees in both Chow rings.
So $\phi$ induces a ring isomorphism
$A_{orb}^{*}(\mathcal{X}(\mathbf{\Sigma_{\theta}}))\cong A_{orb}^{*}(\mathcal{M}(\mathcal{A}))$.
\end{pf}

\begin{rmk}
The presentation (\ref{chowringhyper}) of orbifold Chow ring only depends on the matroid complex corresponding to the map
$\beta: \mathbb{Z}^{m}\rightarrow N$, not $\theta$. Note that the presentation (\ref{chowringlawrence}) depends on
the fan $\Sigma_{\theta}$. We couldn't see explicitly from this presentation that the ring is independent to the
choice of generic elements $\theta$.
\end{rmk}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to generate routes based on file tree (like Nuxt.js) for vue-router? Nuxt.js generates its routes based on file tree.
Snnipets from the official page:
pages/
--| _slug/
-----| comments.vue
-----| index.vue
--| users/
-----| _id.vue
--| index.vue

From which Nuxt.js generates:
router: {
  routes: [
    {
      name: 'index',
      path: '/',
      component: 'pages/index.vue'
    },
    {
      name: 'users-id',
      path: '/users/:id?',
      component: 'pages/users/_id.vue'
    },
    {
      name: 'slug',
      path: '/:slug',
      component: 'pages/_slug/index.vue'
    },
    {
      name: 'slug-comments',
      path: '/:slug/comments',
      component: 'pages/_slug/comments.vue'
    }
  ]
}

I think this feature is really cool (in fact it's pretty common for standard MVC frameworks, as you know). Without this, managing the routes and components quickly becomes a mess as the project grows.
Is it possible to make this work without Nuxt.js, or specifically in my case, when using vue-cli webpack template?

redpajama_set_name: RedPajamaStackExchange


text:
72% say 'ok' sex scandals
WEB EDITION :: Local News
98% say 'no' to corrupt leaders holding top govt posts
Hemananthani Sivanandam
Tan Sri Ramon V. Navaratnam
KUALA LUMPUR (Feb 3, 2009) : A whopping 98% of the people will not accept politicians involved in corruption holding a senior government post.
This was the finding of the Merdeka Centre for Opinion Research from a voter survey conducted between Dec 26, 2008 and Jan 2 this year.
The survey interviewed 1,018 randomly selected registered voters via telephone.
Transparency International Malaysia president Tan Sri Ramon V. Navaratnam said the findings confirmed TI-Malaysia's perception that corruption is widespread in the country.
"This is borne out by the CPI (Corruption Perception Index) among other countries. Although the government has taken considerable steps to fight corruption, the negative perceptions will continue for several reasons," he said.
Malaysia's ranking on the CPI declined to 47 out of 180 countries in 2008, compared with 36 out of 91 countries in 2001 and 43 out of 179 countries in 2007.
Navaratnam said perceptions are hard to change in the short term, and public perception is that all measures have so far not been able to focus on the major culprits (perpetrators of corruption).
"The Merdeka survey hits the nail on the head and dismisses allegations that the CPI is not accurate enough. Unless more big fishes are caught, the perception will continue," he added.
The survey also asked respondents for their views on the newly formed Malaysian Anti-Corruption Commission (MACC) to replace the Anti-Corruption Agency (ACA).
However, 57% of the Chinese and 50% of the Indian respondents said they were not confident of the commission.
Navaratnam said it is too soon to make a decision as the commission is in its early days. However, he urged the MACC to show it is different from ACA within the first 100 days.
"I urge the MACC to go to the root of all evil and charge and convict some big fish in order to show the difference, or else the public will regard MACC as 'old hat' and this is most unfortunate because opportunity given to MACC will be lost," he said.
Navaratnam said he was confident of better results and performance from the MACC.
The survey also revealed that people can accept politicians involved in sex scandals appointed to senior government positions. About 72% of the respondents said they would not mind of leaders whose career has been tainted by such controversies.
This entry was posted in Uncategorized on February 4, 2009 by datukgoh.
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