text:
Ask an expertCallback
Trademark counseling and prosecution
Design prosecution
Plant varieties protection
Software and databases
Design litigation
Patent annuities
Trademark renewals
Copyright prosecution and litigation
Inventions and utility models
Selection achievements
Technology transfer & License agreements
Computer software & databases
Appellations of origin
Russian regions
Legislation News
We represent half of the 12 largest corporations in the Fortune Global 500 list
We represent half of the 10 leading global innovators (PCT filers)
We represent 4 of the 10 holders of the world's most valuable brands
MainCompany NewsSeptember 8, 2010 the Moscow Arbitration Court took a Decision in favor of Sojuzpatent's client Fo Shan Shi Nan Hai Li Feng Ji Xie Corporation (China) (Plaintiff) regarding the case of infringement of exclusive rights for the trademark under international registration No 936586. By the said Decision it is prohibited for the Defendant to offer for sale and to sell goods, namely, metal-working machines (07 class of ICGS), which are marked by the designation «ACL» confusingly similar to the trademark under international registration No 936586, the Defendant is forbidden to use the designation «ACL» confusingly similar to the trademark under international registration No 936586 on the documentation connected with the introduction of goods, namely, metal-working machines (07 class of ICGS), into the civil turnover in the territory of the Russian Federation, it was also decided to reimburse from the Defendant in favor of Fo Shan Shi Nan Hai Li Feng Ji Xie Corporation compensation for the infringement of trademark rights under the international registration No 936586 in the amount of 100 000 rubles.
September 8, 2010 the Moscow Arbitration Court took a Decision in favor of Sojuzpatent's client Fo Shan Shi Nan Hai Li Feng Ji Xie Corporation (China) (Plaintiff) regarding the case of infringement of exclusive rights for the trademark under international registration No 936586. By the said Decision it is prohibited for the Defendant to offer for sale and to sell goods, namely, metal-working machines (07 class of ICGS), which are marked by the designation «ACL» confusingly similar to the trademark under international registration No 936586, the Defendant is forbidden to use the designation «ACL» confusingly similar to the trademark under international registration No 936586 on the documentation connected with the introduction of goods, namely, metal-working machines (07 class of ICGS), into the civil turnover in the territory of the Russian Federation, it was also decided to reimburse from the Defendant in favor of Fo Shan Shi Nan Hai Li Feng Ji Xie Corporation compensation for the infringement of trademark rights under the international registration No 936586 in the amount of 100 000 rubles.
World Trademark Review:
"A well-established player with a stellar reputation... A top choice for filing and prosecution."
Managing Intellectual Property:
"Sojuzpatent excels in both patent prosecution and contentious work"
IAM Patent 1000:
"Oldest IP law firm in the territory of the former Soviet Union, Sojuzpatent has an enviable position on the Russian patent landscape"
Chambers & Partners, The Client's Guide:
"A terrific firm with many excellent specialists"
Address 13 Bldg. 5 Myasnitskaya st., 101000 Moscow, Russia
Telephone+7 800 555 42 53+7 495 221 88 80+7 495 221 88 81
Fax+7 495 221 88 85+7 495 221 88 86
E-mailinfo@sojuzpatent.com
I certify that I have read and agree to the Privacy Policy*
© Site development and design
InfoDesign, 2011—2019
© Law firm Sojuzpatent Ltd., 2018.
The years of foundation of Sojuzpatent coincided with the Golden Age of the Russian Avant-Garde Art. That is why we used in our web-site design some paintings of this time period—to convey the spirit of the epoch. Sojuzpatent expresses its profound gratitude to the State Tretyakov Gallery, Moscow, for affording it the opportunity to use the following paintings by Aristarkh Lentulov from the Gallery's collection:
1. St. Basil's Cathedral; 2. Ringing. Ivan The Great Bell Tower; 3. Gate with a tower. The New Jerusalem; 4. Tverskoy Boulevard.


Once you have reached Rome outer limits, take the ring road "GRA-Grande Raccordo Anulare". Follow signs to and exit Roma Centro/Vatican City/Aurelia (Exit 1). Keep going straight on Via Aurelia and follow signs to Piazza San Giovanni Battista della Salle and Piazza di Villa Carpegna. Once you get there, turn left on Via della Madonna del Riposo, keep going straight, following signs to Ospedale San Carlo di Nancy (Hospital) on Via Aurelia. You will soon reach Via Aurelia, 190.
Alternatively you can take the train that connects the airport with the city.
Get off at the Trastevere stop and change to the FM3 line towards San Pietro. Get off at the San Pietro station, exit at Piazzale Gregorio VII, and take via Aurelia. Follow it for about 300 m, until you reach number 190.
There are several connections between the airport and Ciampino station, from where you can arrive by train to Termini Station in Rome.
Alternatively, take the taxi that reaches Vatican Home in about 40 minutes.
If you wish, we could send you to take a driver who will meet you at the ``Arrivals`` area showing a sheet that says your name and surname.
Take the subway line A with direction Battistini. Get off at the Cipro-Musei Vaticani station exit towards the Vatican Museums. Climb the stairs of via Vittor Pisani and turn right onto viale Vaticano. Continue for 400 m, until you cross via Aurelia. Go left on via Aurelia and after 50 m on the left you will find street number 190, or from viale Vaticano take bus 49, get off at Porta Pertusa, in front of the downhill road stop, third building on the left.
Alternatively, take a taxi: 15 minutes on average with an average cost of € 12.
Take the underground line B towards Laurentina and get off at Termini station. Change to underground line A towards Battistini and get off at Cipro - Musei Vaticani stop. Climb the stairs of via Vittor Pisani and turn right onto viale Vaticano. Continue for 400 m, until you cross via Aurelia. Get off on via Aurelia, on the left, and after 50 m on the left is the street number 190, or take bus 49, get off at Porta Pertusa, in front of the downhill road stop, the third building on the left.
Alternatively: take FR1 Regional Railways towards Piramide, get off at Piramide station and take the FR5 line towards San Pietro, get off at San Pietro station, exit at Piazzale Gregorio VII, and take via Aurelia. Follow it for about 300 m, until you reach number 190.


text:
Poodle Conquers Three-Year Battle with Disease
Penny was treated for immune mediated thrombocytopenia over three years at the UC Davis veterinary hospital.
Posted by Rob Warren
"Case of the Month" – October 2019
Penny, a 7-year-old female poodle mix, first came to the UC Davis veterinary hospital in 2016. Through more than 100 visits to the hospital over the past three years, pet parents Amanda Frew and Amanda Brown have helped Penny battle immune mediated thrombocytopenia – a condition that caused Penny's immune system to attack and destroy her own blood platelets. Without platelets, Penny's blood would be unable to clot properly, putting her at risk of bleeding to death with even minor injuries.
Most cases of this disease respond well to immunosuppression alone, sometimes with only a single medication or occasionally needing more than one. When Penny initially presented to UC Davis, she was on three different medications. Drs. Krystle Reagan and Laura Lim, residents with the Internal Medicine Service, initially tried a specific kind of immunosuppressant agent from human blood that helps slow down an overzealous immune system and has been shown to help some dogs with immune mediated thrombocytopenia. Penny responded well initially, but unfortunately relapsed.
Because the spleen is where much platelet destruction occurs in dogs with immune mediated thrombocytopenia, Penny's spleen was removed, and she began to respond well and had her medications tapered. However, Penny continued to respond and relapse to medications two more times throughout 2018. A relapse at the beginning of 2019 became much more complicated.
By this time, Drs. Lim and Reagan completed their residencies (a three-year advanced training program to become a specialist), and resident Dr. Craig Sutter joined Penny's care team. She was switched from prednisone to cyclosporine, and the veterinarians started a new full diagnostic work-up to continue to evaluate for Penny's elusive underlying cause.
When an abdominal ultrasound was repeated, the team found a large blood clot in one of the main veins leading to her liver. This blood clot was likely formed from some combination of the two medications she was on for her disease. Most times, patients in this condition are administered anti-clotting medications, but Penny was too high risk for this, as the main side effect of anti-clotting medications is bleeding. If Penny started to bleed, she would not be able to stop.
Penny's immunosuppressive medications were switched in an effort to better control her disease and improve her platelet count. However, she began to bleed into her gastrointestinal tract from her underlying disease and became severely ill. While Drs. Reagan and Sutter contemplated discussing quality of life with Frew and Brown, Penny continued to fight.
"Never at any point did it appear that her spirit diminished," said Frew. "We couldn't give up on her because she was never going to give up on herself."
She responded well while hospitalized and although her platelets didn't increase, she stopped bleeding. The team switched her immunosuppressive medications once again to a combination of drugs they had never used before in an effort to control her particular disease. This last combination finally turned out to be what Penny needed. Her platelets improved to the highest levels in years, increasing to a normal level where she could start on an anti-clotting medication to break down the massive clot discovered on the ultrasound. At a recent appointment, Penny's care team reported that only a small fragment of the clot remains.
"Penny has suffered rare but severe complications from her multiple immunosuppressive medications – complications that would cause most animals to pass," Dr. Sutter said. "Both Dr. Reagan and I would have given Penny a poor prognosis not even two months ago, and now she is doing so well and continues to have normal platelet levels."
"All the doctors, students and especially the techs—we know the techs better than anyone—have all been fantastic," said Frew. "It's great that the whole team knows Penny so well. Every time we came, everyone in the hallway stopped and said hello to Penny."
Throughout Penny's years of treatments, she had to receive multiple blood transfusions. Thankfully, the UC Davis veterinary hospital runs the largest veterinary blood bank on the West Coast. Every year, hundreds of dogs from the community donate more than 600 pints of blood. One dog's donations can help up to five dogs like Penny who are hospitalized at UC Davis.
"I don't know that any other place in the world could've been able to handle this case," continued Frew, discussing the complexity of Penny's disease. "This was a very complicated case with a lot of complicated terminology, so it was really nice to have doctors explain things in a way we could understand. They were so thorough in their work-ups of Penny's case. They thought of every potential scenario and every possible way to treat it."
"This was a case unlike any I've seen," said Dr. Sutter. "I felt like we had to try everything for Penny."
"You can tell that everyone there is equally as apt at their job as you could hope for," Frew said. "The level of competency is unmatched."
Across the board—from the veterinary team, to the owners, to Penny herself—a dedicated, never-give-up effort over three years gave a little dog her life back.
"Now she is more energetic and vibrant than ever," said Frew. "She's a happy, healthy little girl again."
See more photos of Penny
Case of the Month internal medicine Dogs

redpajama_set_name: RedPajamaCommonCrawl


— Humanity is devouring our planet's resources in increasingly destructive volumes, according to a new study that reveals we have consumed a year's worth of carbon, food, water, fibre, land and timber in a record 212 days.
As a result, the Earth Overshoot Day – which marks the point at which consumption exceeds the capacity of nature to regenerate – has moved forward two days to 1 August, the earliest date ever recorded.
To maintain our current appetite for resources, we would need the equivalent of 1.7 Earths, according to Global Footprint Network, an international research organisation that makes an annual assessment of how far humankind is falling into ecological debt.
The overshoot began in the 1970s, when rising populations and increasing average demands pushed consumption beyond a sustainable level. Since then, the day at which humanity has busted its annual planetary budget has moved forward.
Thirty years ago, the overshoot was on 15 October. Twenty years ago, 30 September. Ten years ago, 15 August. There was a brief slowdown, but the pace has picked back up in the past two years. On current trends, next year could mark the first time, the planet's budget is busted in July.
While ever greater food production, mineral extraction, forest clearance and fossil-fuel burning bring short-term (and unequally distributed) lifestyle gains, the long-term consequences are increasingly apparent in terms of soil erosion, water shortages and climate disruption.
The situation is reversible. Research by the group indicates political action is far more effective than individual choices. It notes, for example, that replacing 50% of meat consumption with a vegetarian diet would push back the overshoot date by five days. Efficiency improvements in building and industry could make a difference of three weeks, and a 50% reduction of the carbon component of the footprint would give an extra three months of breathing space.
In the past, economic slowdowns – which tend to reduce energy consumption – have also shifted the ecological budget in a positive direction. The 2007-08 financial crisis saw the date push back by five days. Recessions in the 90s and 80s also lifted some of the pressure, as did the oil shock of the mid 1970s.
But the overall trend is of costs increasingly being paid by planetary support systems.


2007 Volvo S40 Review - I realize that you're seeking articles on our blog within the headline 2007 Volvo S40 Review of choices of articles that we got. If you are trying to find 2007 Volvo S40 Review article and you fails to dwell right here, you can see from several of the articles that may suit your needs right down below. We are going to help you to get articles 2007 Volvo S40 Review you're looking at some of our content which might be suitable as you are looking for the material. Thank you for coming to our website. We really hope that you will find what you want to find here related to the 2007 Volvo S40 Review.
2007 Volvo S40 Review - Used 2007 volvo s40 consumer reviews - 45 car reviews, I purchased my 2007 volvo s40 in late 2007. the vehicle had less than 5,000 miles and was in perfect condition. i have always been an import buyer yet never owning a volvo i was a little skeptical.. 2007 volvo s40 consumer reviews | cars., 5 out of 5 stars. best volvo for the money! by volvo driver for life from augusta ga on sat feb 10 2007 as a volvo driver and owner for more than 20 years, i have been sold on the safety of these. 2007 volvo s40 reviews rating | motortrend, Motor trend reviews the 2007 volvo s40 where consumers can find detailed information on specs, fuel economy, transmission and safety. find local 2007 volvo s40 prices online.. 2007 volvo s40 - user reviews - cargurus, 2007 volvo s40 reviews: read 7 candid owner reviews for the 2007 volvo s40. get the real truth from owners like you.. 2007 volvo s40 | pricing, ratings & reviews - kbb., 2007 volvo s40 kbb expert review. the s40 is a fantastic way to enjoy the perks of owning a premium brand automobile without having to pay a premium price. for about the same money as a honda. 2007 volvo s40 expert reviews, specs photos | cars., Research the 2007 volvo s40 online at cars.com. you'll find local deals, specs, images, videos, consumer and expert reviews, features, trims, and articles for every turn in your s40..


There is no standard vaccination program for all horses. We can evaluate the risk of disease based on anticipated exposure, environmental and geographic factors, and the age, breed, and use of your horse. Good management practices are essential in addition to vaccination in order to prevent infectious diseases. We are happy to design a vaccine program tailored to your situation. Core vaccines are those vaccines that every horse in the United States should have. These vaccines include Eastern & Western Encephalomyelitis, Tetanus Toxoid, West Nile Virus, and Rabies. Risk based vaccines are those vaccines recommended based on risk of exposure and environment. Some areas of the country are endemic for certain diseases. Risk based vaccines include Flu (Influenza) and Rhinopneumonitis (Herpesvirus), Potomac Horse Fever, Strangles, and Botulism.


text:
Guilds reappraised: Italy in the early modern period
The article presents the main results of a collective research programme of national interest carried out between 1997 and 2002 and encouraged by the desire critically to reconsider Cipolla's view that guilds were one of the main reasons for the decline of the Italian economy during the seventeenth century. The Italian research group decided to support specific research on situations in individual towns by setting up a database, which has made it possible to substantiate many of the claims that were emerging about the function and history of guilds. The article, starting from database informations, contributes to international debate on guilds and provides new insights on Italian case with special regards to the relationship between guilds and economic trend, guilds and technical progress, guilds and apprenticeship. In particular it emphasized that the dynamic role played by guilds did not diminish, but rather was redefined to highlight both their mercantile function as new markets began to develop and their ability to coordinate decentralized production in a pre-industrial environment. It seems necessary to clear urban guilds of the accusation that they were responsible for a lengthy crisis since they were crucial in a deeper and quite well-defined process of reorganization.
Mocarelli, L. (2008). Guilds reappraised: Italy in the early modern period. INTERNATIONAL REVIEW OF SOCIAL HISTORY, 53(supplement 16), 159-178.
Citazione: Mocarelli, L. (2008). Guilds reappraised: Italy in the early modern period. INTERNATIONAL REVIEW OF SOCIAL HISTORY, 53(supplement 16), 159-178.
Titolo: Guilds reappraised: Italy in the early modern period
Autori: Mocarelli, L
MOCARELLI, LUCA
Rivista: INTERNATIONAL REVIEW OF SOCIAL HISTORY
Digital Object Identifier (DOI): http://dx.doi.org/10.1017/S0020859008003659


text:
import os

from setuptools import find_packages
from setuptools import setup

version = '2.1.0-1'

def read(*rnames):
    return open(os.path.join(os.path.dirname(__file__), *rnames)).read()

long_description = (read('README.txt') + '\n' +
                    read('js', 'raphael', 'test_raphael.txt') + '\n' +
                    read('CHANGES.txt'))

setup(
    name='js.raphael',
    version=version,
    description="Fanstatic packaging of Raphael",
    long_description=long_description,
    classifiers=[],
    keywords='',
    author='Andreas Kaiser',
    author_email='disko@binary-punks.com',
    url='https://github.com/disko/js.raphael',
    license='BSD',
    packages=find_packages(),
    namespace_packages=['js'],
    include_package_data=True,
    zip_safe=False,
    install_requires=[
        'fanstatic',
        'setuptools',
    ],
    entry_points={
        'fanstatic.libraries': [
            'raphael = js.raphael:library',
        ],
    },
)

redpajama_set_name: RedPajamaGithub


After (M)onday and (T)uesday comes WTF !
I'd take a nerf bullet for you.
Today I laughed until my abs started hurting, so I can skip the gym.
Hold on, let me overthink this.
I would like to thank my arms for always being by my side, my legs for always supporting me, and my fingers, because I can always count on them.
Christmas is a competition between who gives up first: Your feet or your wallet.
I noticed you are not noticing me.
The word aerobics comes from two Greek words: aero, meaning "ability to," and bics, meaning "withstand tremedous boredom."
Remember: Don't Insult the Alligator till after you cross the river.
I bet you anything that I can stop gambling right now.
A toy that can't be broken can be used to break other toys.
Sorry I'm late, I got here as soon as I wanted to.
The worst part of online shopping is having to get up and get your card out of your wallet.
A misty day does not signify a cloudy day, it signifies frizzy hair.
Displayed 201-225 of 236 quotes.


text:
package main

import (
	"fmt"
	"io/ioutil"
	"log"
	"path"

	"github.com/yannh/r10k-go/git"
	"github.com/yannh/r10k-go/puppetsource"
)

type environment struct {
	source        puppetsource.Source
	branch        string
	modulesFolder string
}

func newEnvironment(s puppetsource.Source, branch string) environment {
	return environment{
		s, branch, "modules",
	}
}

func getEnvironments(envNames []string, sources []puppetsource.Source) []environment {
	envs := make([]environment, 0)

	for _, envName := range envNames {
		// Find in which source the environment is
		// TODO: make deterministic
		found := false
		for _, source := range sources {
			if git.RepoHasRemoteBranch(source.Remote(), envName) {
				envs = append(envs, newEnvironment(source, envName))
				found = true
				break
			}
		}
		if found == false {
			log.Printf("failed to find source for environment %s", envName)
		}
	}

	fmt.Printf("%+v\n", envs)
	return envs
}

func (e *environment) installedModules() []string {
	folder := path.Join(e.source.Basedir(), e.branch, e.modulesFolder)

	files, err := ioutil.ReadDir(folder)
	if err != nil {
		log.Fatal(err)
	}

	modules := make([]string, 5)
	for _, f := range files {
		modules = append(modules, f.Name())
	}

	return modules
}

func (env *environment) fetch(cache *cache) error {
	s := env.source.(*puppetsource.GitSource) // FIXME should use interface instead of checkout/clone
	if err := s.Fetch(cache.folder); err != nil {
		return err
	}

	if err := git.Checkout(s.Location(), git.NewRef(git.TypeBranch, env.branch)); err != nil {
		return err
	}
	if err := git.Clone(s.Location(), path.Join(s.Basedir(), env.branch)); err != nil {
		return err
	}

	return nil
}

func DeployedEnvironments(s puppetsource.Source) []environment {
	folder := path.Join(s.Basedir())

	files, err := ioutil.ReadDir(folder)
	if err != nil {
		log.Fatal(err)
	}

	envs := make([]environment, 0)

	for _, f := range files {
		envs = append(envs, newEnvironment(s, f.Name()))
	}

	return envs
}


*This jacket and Muggle clothes is one possible outfit I am considering for the premiere of The Force Awakens. I'm in at 8 PM Thursday, December 17 (thirty-one days!). Others are: Full Jedi, Yavin Luke, or tying to get Death Star Duel Luke finished on time. Decisions, decisions...funny how I've been thinking on what to wear to a movie for weeks and didn't give two cents about my wedding. Oh. Wait. That's not odd at all. Priorities, man. Priorities.
Not familiar with Star Wars much, but that's one awesome looking jacket! Congrats!
I do plan on seeing the new. Let me know how you like it!


As the first international lighting fair in Central America and the Carribean Region, Interlumi Panama 2016 witnessed a complete success, receiving over 120 lighting brands from 18 countries/regions and almost 2,000 visitors from 27 countries. The exhibitors praised the professionalism of InterLumi Panama and its high quality visitors. 87% of them showed interest in participating in the Interlumi 2017 edition as well. Visitors, on it's turn, were impressed by a wide variety of lighting products on site.
The next edition, Interlumi 2017, will keep up with the trend of the global Lighting industry. Aiming to become an international platform for exibitors, Inter-Lumi Panama 2017 would like to involve more participants from all over the world to light up the path to the Latin American & Carribean market.
In addition to hosting a variety of exhibitors and products on display, the show also holds a concurrent event to help participants enrich their knowledge and add value to their visits: The Efficient Lighting Forum.
Debuted alongside the first InterLumi, the forum ushered the audience into the amazing world of the lighting industry. 10 professional speakers were invited to share ideas and expertise, providing an informative platform for the participants to learn more about the practical experience as well as the application of state-of-the-art technologies in the market.
Amid applause and high recognition, the forum in 2017 will further in-depth communication to inspire the region's lighting practitioners to discover the prospect of efficient lighting in the LATAM area. With simultaneous interpretation provided, topics featuring LED lighting, smart lighting, renewable energy, novel materials and eco-friendly lighting design will be discussed without any language barrier.
Visit the official event website at http://www.inter-lumi.com.
InterLumi Panama, the global lighting trade show organized by America Expo Group, is running on an annual basis at the ATLAPA Convention Center in Panama. Serving the Latin American & the Caribbean countries, InterLumi Panama is positioned to be an effective tool to stimulate the development of the region's lighting industry as well as provide a gateway for exhibitors to tap into this emerging market.


text:
Teachers Make Less than Deserved
katherinelcox
A baseball player sits on the bench, chewing gum and watching his team take down the opposing team on the diamond. Sitting on the bench isn't too bad in his opinion. No stress, no work, and still over $1 million rolling in to allow him to enjoy the finer things in life.
Meanwhile, a teacher sits in her classroom, running her fingers through her hair. After hours of teaching young children, she slaves over the lesson plans for the next week. Her students are struggling with a concept, and she needs a new approach. The stress continues to build, and outside the classroom, things aren't much better as she worries about how she is going to pay for her own kids' clothes and sports this season.
The difference between the salaries of teachers and athletes is astounding. According to Yahoo! Sports, the average salary of NBA players is about $5.15 million per year. This is only an average, and even benchwarmers make millions of dollars easily. The average salary of a teacher in 2010/11, on the other hand, was $56,069, according to the Department of Education.
I can't think of any better way to say this but to call it ridiculous. How can teachers, who educate the future generations of our country, make so little, while athletes, who do nothing but play a game, make so much?
Even a player who sits on the bench can make plenty of money to comfortably support a family, while being able to buy extra things as well.
Many teachers make so little that they struggle to support their family. They must save every penny they make to put food on the table.
According to the National Science Foundation, over half of public math and science teachers were dissatisfied with their salary. The numbers increase for teachers working in high poverty districts.
Teachers are some of the most important people in our society. Without teachers, how would children learn to be productive and able adults who will benefit our country? People who go into teaching know that they won't make much money, and they still choose it as a career because they care. These teachers deserve a reward. They deserve a thank you from everyone because without teachers, none of us would be the people we are today.
What do we have to thank athletes for? Sure, they provide us some entertainment, and sure, their strength and talent is admirable, but beyond entertainment they don't really contribute much to our society with those talents.
People say that athletes make a lot of money through ticket sales, which is true. But does it really seem fair that all the ticket sales money goes back to the athletes, even those on the bench, when teachers work every day and make barely enough money to get by? Perhaps there could be a separate tax that goes specifically to teachers' salaries, or maybe they could even take a percentage of excess profit from the sports domain and give it to the teaching domain. Our future generations depend on teachers, and we have all received the benefits of teachers at some point. Teachers deserve a much bigger salary than they currently receive, especially when athletes make so much.
K through 12
All Knight Life Picks Reader Picks Sort: Newest


text:
Q: template with enum specialization I have a template which takes as specialization an enum class (note the different initializations of enum), and I would like to be able to compare two different enums
enum choiceA {One, Two, Three};
typedef enum  {Four, Five, Six} choiceB;

template<typename S>
void f(S choice, int& x)
{
if (choice == One || choice == Four) { x = SomeWhateverIntegerValue;}
else if(choice = Five) {x = AnOtherWhateverIntegerValue;}
else {x = OtherIntegerValue;}
}

so that
int x = 0;
f<choiceA>(One, x);

works as intended.

A: The fact that you have two enum types for the same enumeration is odd. I would fix that design first.
Anyway to strictly answer your question you don't need templates here. A simple overload would suffice:

void f(choiceA choice, int& x)
{
    if (choice == One)
    {
        x = SomeWhateverIntegerValue;
    }
    else
    {
        x = OtherIntegerValue;
    }
}

void f(choiceB choice, int& x)
{
    if (choice == Four)
    {
        x = SomeWhateverIntegerValue;
    } else if(choice == Five)
    {
        x = AnOtherWhateverIntegerValue;
    }
    else
    {
        x = OtherIntegerValue;
    }
}

As a side note avoid the C style typedef enum/class { ... } Name;.
Also take care if(choice = Five) you have a dangerous typo. It should be ==. Enable compiler warnings, most modern compilers warn about this.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Why two different keyboard connected to a single computer have telepathy? I connected two different keyboard to my computer. (Both use USB port to communicate with the computer.) The question is:
Why when I press Caps-Lock or Num-Lock in one keyboard, The related light (The LED on the keyboard) on the other one turns on and turns off also!?
As @Tetsujin mentioned in the first comment, I guess the computer receive the request from one keyboard, and send the answer broadcasting to all ports.If so, why? Doesn't have each port a specific address to send answer just for that address? Doesn't this current mechanism increase vulnerability? (For example a hardware keylogger)

A: Windows only accepts one keyboard at a time (without specialized driver software), so the interactions with one keyboard affect all keyboards.  Although you can plug multiple keyboards in and the all work, they are all dealt with through the same single interface in the OS, and are all treated equally.
The keyboards themselves only send key-codes to Windows tell it which key is pressed, all keyboards send the same key-codes (in general), and it's up to the OS (Windows) to decide what to do with them.  Because of this, the OS actually controls the state of the modifier keys like Caps and Num.
Since all attached keyboards are affected the same, the OS updates all keyboards to be in the same state when Caps lock (etc.) is toggled, regardless of which Caps lock key was used.
You can expand on the effect by turning on the Windows onscreen keyboard. You'll note its visual key-presses reflect the modifier keys (shift, control, alt, Caps lock, etc.) you press on your physical keyboard(s).
For further reading, perhaps check out these other SU questions:

*

*Why can't I use two or more keyboards/mice at the same time on one computer?

*Using multiple keyboards with different keyboard layouts in Windows


text:
GLOBAL MEDICAL REIT ANNOUNCES SECOND QUARTER 2021 FINANCIAL RESULTS
BETHESDA, Md.--(BUSINESS WIRE)-- Global Medical REIT Inc. (NYSE: GMRE) (the "Company" or "GMRE"), a net-lease medical office real estate investment trust (REIT) that owns and acquires purpose-built healthcare facilities and leases those facilities to strong healthcare systems and groups with leading market share, today announced financial results for the three and six months ended June 30, 2021 and other data.
Second Quarter 2021 Highlights
Net income attributable to common stockholders was $2.6 million, or $0.04 per diluted share, as compared to $0.2 million, or $0.00 per diluted share, in the comparable prior year period.
Funds from Operations ("FFO") of $0.22 per share and unit, as compared to $0.19 per share and unit in the comparable prior year period.
Adjusted Funds from Operations ("AFFO") of $0.23 per share and unit, as compared to $0.21 per share and unit in the comparable prior year period.
Increased total revenue 28.1% year-over-year to $28.3 million, primarily driven by the Company's acquisition activity.
Completed seven acquisitions for an aggregate purchase price of $71 million and a weighted average cap rate of 7.2%.
Generated approximately $51 million in gross proceeds from "at-the-market" ("ATM") equity offerings at an average price of $15.27 per share, maintaining leverage at 41% as of June 30, 2021.
As previously announced, on May 3, 2021, amended and restated credit facility to expand capacity, reduce credit spreads, convert to unsecured, and extend maturity.
Six Month and Other 2021 Highlights
FFO of $0.44 per share and unit, as compared to $0.38 per share and unit in the comparable prior year period.
AFFO of $0.47 per share and unit, as compared to $0.41 per share and unit in the comparable prior year period.
Completed 11 acquisitions through June 30, 2021, for an aggregate purchase price of $113.8 million at a weighted average cap rate of 7.4%. Subsequent to quarter end, completed an additional two acquisitions for an aggregate purchase price of $26.2 million at a weighted average cap rate of 7.4%.
Through June 30, 2021, generated approximately $201 million in gross proceeds from equity offerings.
Jeffrey M. Busch, Chairman, Chief Executive Officer and President stated, "Our strong performance in the first half of the year can be attributed to the focused efforts and hard work of our team. During the quarter, we produced 28% revenue growth year-over-year, with the portfolio 99% occupied, and a Rent Coverage Ratio of 4.4x. Additionally, during the quarter we amended our credit facility, both increasing available capacity and improving our cost of capital, providing significant liquidity to support continued accretive growth. Though the market remains competitive, we remain confident in our ability to source and acquire high quality medical facilities at attractive yields. We look forward to the balance of the year with confidence as we continue to grow cash flow and create value for our stockholders."
Rental revenue for the second quarter of 2021 increased 28.0% year-over-year to $28.2 million, reflecting the growth in the Company's portfolio.
Total expenses for the second quarter were $24.1 million, compared to $20.4 million for the comparable prior year period, primarily reflecting the growth in the Company's property portfolio. Interest expense for the second quarter was $5.0 million, compared to $4.4 million for the comparable prior year period. This increase is primarily due to higher average borrowings during the quarter which helped fund our property acquisitions.
Net income attributable to common stockholders for the second quarter totaled $2.6 million, or $0.04 per diluted share, compared to $0.2 million, or $0.00 per diluted share, in the comparable prior year period.
The Company reported FFO of $0.22 per share and unit for the second quarter, as compared to $0.19 per share and unit in the comparable prior year period. AFFO was $0.23 per share and unit for the second quarter versus $0.21 per share and unit in the comparable prior year period.
Acquisitions Update
During the second quarter of 2021, the Company completed seven acquisitions, encompassing an aggregate 234,173 leasable square feet, for an aggregate purchase price of $71 million. The properties were purchased at a 7.2% weighted average cap rate.
Since July 1, 2021, the Company completed two acquisitions encompassing 77,693 leasable square feet for an aggregate purchase price of $26.2 million and a weighted average cap rate of 7.4%. To date, the Company has completed 13 acquisitions in 2021, encompassing 432,122 leasable square feet for an aggregate purchase price of $140.0 million and a weighted average cap rate of 7.4%.
Additionally, the Company has three properties under contract for an aggregate purchase price of $23.2 million. The properties are currently in the due diligence period and we can make no assurances that the acquisitions will occur on a timely basis if at all.
As of June 30, 2021, the Company's portfolio was 99.1% occupied and comprised of 4.1 million leasable square feet with an annualized base rent of $96.8 million. The weighted average lease term for the Company's portfolio was 7.5 years and the weighted average annual rental escalations for the portfolio was 2.1%. As of June 30, 2021, the Company's portfolio Rent Coverage Ratio (as defined below) was 4.4 times.
During the three and six months ended June 30, 2021, the Company issued 3.4 million and 14.7 million shares of its common stock, respectively, generating gross proceeds of $51.2 million and $201.3 million, respectively. These issuances include:
an underwritten offering of 8.6 million shares at an offering price of $13.30 per share that generated gross proceeds of $114.7 million that closed on March 18, 2021, and
6.1 million shares issued through our ATM offering program that generated gross proceeds of $86.6 million. Our first quarter ATM issuances totaled 2.7 million shares at an average offering price of $13.07 per share, for gross proceeds of $35.4 million, and our second quarter ATM issuances totaled 3.4 million shares at an average offering price of $15.27 per share, for gross proceeds of $51.2 million.
As a result of these equity issuances the Company reduced its leverage from 52% at December 31, 2020 to 41% at both March 31 and June 30, 2021.
As previously announced, on May 3, 2021, the Company amended and restated its credit facility to, among other things, (i) increase the facility's overall capacity by $150 million to $750 million, consisting of a $400 million revolver component and a $350 million term loan component, (ii) reduce its borrowing costs across its pricing grid, (iii) convert to an unsecured facility, and (iv) extend the initial maturity date of the revolver component of the facility to four years (with two, six-month extension options) and the maturity date of the term loan component to five years. The amended and restated credit facility also contains a $500 million accordion feature.
At June 30, 2021, total debt outstanding, including outstanding borrowings on the Credit Facility and notes payable (both net of unamortized debt issuance costs), was $506.8 million. As of June 30, 2021, the Company's debt carried a weighted average interest rate of 3.09% and a weighted average remaining term of 4.71 years. As of August 2, 2021, the Company's unutilized borrowing capacity under the revolver was approximately $265 million.
On June 10, 2021, the Board of Directors declared a $0.205 per share cash dividend to common stockholders of record as of June 24, 2021, which was paid on July 8, 2021, representing the Company's second quarter 2021 dividend payment to its common stockholders. The Board also declared a $0.46875 per share cash dividend to holders of record as of July 15, 2021 of its Series A Preferred Stock, which was paid on August 2, 2021. This dividend represented the Company's quarterly dividend on its Series A Preferred Stock for the period from April 30, 2021 through July 30, 2021.
Details regarding these results can be found in the Company's supplemental financial package available on the Investor Relations section of the Company's website at http://investors.globalmedicalreit.com/.
CONFERENCE CALL AND WEBCAST INFORMATION
The Company will host a live webcast and conference call on Wednesday, August 4, 2021 at 9:00 a.m. Eastern Time. The webcast is located on the "Investor Relations" section of the Company's website at http://investors.globalmedicalreit.com/.
To Participate via Telephone:
Dial in at least five minutes prior to start time and reference Global Medical REIT Inc.
Domestic: 1-877-705-6003
Replay:
An audio replay of the conference call will be posted on the Company's website.
ABOUT GLOBAL MEDICAL REIT
Global Medical REIT Inc. is net-lease medical office REIT that acquires purpose-built specialized healthcare facilities and leases those facilities to strong healthcare systems and physician groups with leading market share.
FFO and AFFO are non-GAAP financial measures within the meaning of the rules of the United States Securities and Exchange Commission ("SEC"). The Company considers FFO and AFFO to be important supplemental measures of its operating performance and believes FFO is frequently used by securities analysts, investors, and other interested parties in the evaluation of REITs, many of which present FFO when reporting their results. In accordance with the National Association of Real Estate Investment Trusts' ("NAREIT") definition, FFO means net income or loss computed in accordance with GAAP before non-controlling interests of holders of OP units and LTIP units, excluding gains (or losses) from sales of property and extraordinary items, less preferred stock dividends, plus real estate-related depreciation and amortization (excluding amortization of debt issuance costs and above and below market lease amortization expense), and after adjustments for unconsolidated partnerships and joint ventures. Because FFO excludes real estate-related depreciation and amortization (other than amortization of debt issuance costs and above and below market lease amortization expense), the Company believes that FFO provides a performance measure that, when compared period-over-period, reflects the impact to operations from trends in occupancy rates, rental rates, operating costs, development activities and interest costs, providing perspective not immediately apparent from the closest GAAP measurement, net income or loss.
AFFO is a non-GAAP measure used by many investors and analysts to measure a real estate company's operating performance by removing the effect of items that do not reflect ongoing property operations. Management calculates AFFO by modifying the NAREIT computation of FFO by adjusting it for certain cash and non-cash items and certain recurring and non-recurring items. For the Company these items include: (a) recurring acquisition and disposition costs, (b) loss on the extinguishment of debt, (c) recurring straight line deferred rental revenue, (d) recurring stock-based compensation expense, (e) recurring amortization of above and below market leases, (f) recurring amortization of debt issuance costs, (g) recurring lease commissions, (h) management internalization costs and (i) other items.
Management believes that reporting AFFO in addition to FFO is a useful supplemental measure for the investment community to use when evaluating the operating performance of the Company on a comparative basis. The Company's FFO and AFFO computations may not be comparable to FFO and AFFO reported by other REITs that do not compute FFO in accordance with the NAREIT definition, that interpret the NAREIT definition differently than the Company does, or that compute FFO and AFFO in a different manner.
RENT COVERAGE RATIO
For purposes of calculating our portfolio weighted-average EBITDARM coverage ratio ("Rent Coverage Ratio"), we excluded credit-rated tenants or their subsidiaries for which financial statements were either not available or not sufficiently detailed. These ratios are based on latest available information only. Most tenant financial statements are unaudited and we have not independently verified any tenant financial information (audited or unaudited) and, therefore, we cannot assure you that such information is accurate or complete. Certain other tenants (approximately 11% of our portfolio) are excluded from the calculation due to (i) lack of available financial information or (ii) receipt of significant COVID-19 relief funds that may cause reported coverage to differ materially from underlying performance. Additionally, our Rent Coverage Ratio adds back physician distributions and compensation. Management believes all adjustments are reasonable and necessary.
Certain statements contained herein may be considered "forward-looking statements" within the meaning of the Private Securities Litigation Reform Act of 1995, and it is the Company's intent that any such statements be protected by the safe harbor created thereby. These forward-looking statements are identified by their use of terms and phrases such as "anticipate," "believe," "could," "estimate," "expect," "intend," "may," "should," "plan," "predict," "project," "will," "continue" and other similar terms and phrases, including references to assumptions and forecasts of future results. Except for historical information, the statements set forth herein including, but not limited to, any statements regarding our earnings, expected financial performance (including future cash flows), our ability to source and acquire high quality medical facilities at attractive yields, future dividends or other financial items; any other statements concerning our plans, strategies, objectives and expectations for future operations, our pipeline of acquisition opportunities and expected acquisition activity, including the timing and/or successful completion of any acquisitions and expected rent receipts on these properties, and any statements regarding future economic conditions or performance are forward-looking statements. These forward-looking statements are based on our current expectations, estimates and assumptions and are subject to certain risks and uncertainties. Although the Company believes that the expectations, estimates and assumptions reflected in its forward-looking statements are reasonable, actual results could differ materially from those projected or assumed in any of the Company's forward-looking statements. Additional information concerning us and our business, including additional factors that could materially and adversely affect our financial results, include, without limitation, the risks described under Part I, Item 1A - Risk Factors, in our Annual Report on Form 10-K, our Quarterly Reports on Form 10-Q, and in our other filings with the SEC. You are cautioned not to place undue reliance on forward-looking statements. The Company does not intend, and undertakes no obligation, to update any forward-looking statement.
Global Medical REIT Inc.
(unaudited, and in thousands, except par values)
Investment in real estate:
Acquired lease intangible assets
Less: accumulated depreciation and amortization
Investment in real estate, net
Tenant receivables, net
Due from related parties
Escrow deposits
Deferred assets
Derivative asset
Credit Facility, net of unamortized debt issuance costs of $8,960 and $3,559 at June 30, 2021 and December 31, 2020, respectively
Notes payable, net of unamortized debt issuance costs of $706 and $835 at June 30, 2021 and December 31, 2020, respectively
Accounts payable and accrued expenses
Dividends payable
Derivative liability
Acquired lease intangible liability, net
Preferred stock, $0.001 par value, 10,000 shares authorized; 3,105 issued and outstanding at June 30, 2021 and December 31, 2020, respectively (liquidation preference of $77,625 at June 30, 2021 and December 31, 2020, respectively)
Common stock, $0.001 par value, 500,000 shares authorized; 64,197 shares and 49,461 shares issued and outstanding at June 30, 2021 and December 31, 2020, respectively
Total Global Medical REIT Inc. stockholders' equity
Noncontrolling interest
Total liabilities and equity
(unaudited, and in thousands, except per share amounts)
June 30,
Rental revenue
Management fees – related party
Depreciation expense
Amortization expense
Management internalization expense
Preacquisition expense
Less: Preferred stock dividends
Less: Net income attributable to noncontrolling interest
Net income attributable to common stockholders
Net income attributable to common stockholders per share – basic and diluted
Weighted average shares outstanding – basic and diluted
Reconciliation of Net Income to FFO and AFFO
(unaudited, and in thousands, except per share and unit amounts)
Amortization of above market leases, net
Straight line deferred rental revenue
Amortization of debt issuance costs and other
FFO per share and unit
AFFO per share and unit
Weighted Average Shares and Units Outstanding – basic and diluted
Weighted Average Shares and Units Outstanding:
Weighted Average OP Units
Weighted Average LTIP Units
Stephen Swett
stephen.swett@icrinc.com
Source: Global Medical REIT Inc.
Global Medical REIT, Inc.
Symbol: GMRE
REIT Sector: Health Care
Externally Managed REIT


How Far is Sreevatsa Shivalaya?
Sreevatsa Shivalaya is one of the popular residential developments in Ram Nagar, neighborhood of Coimbatore. It is among the ongoing projects of its Builder. It has lavish yet thoughtfully designed residences in 1 Blocks.


text:
\section{Introduction}

We work over the complex numbers field ${\mathbb C}$.
Let $Q=(Q_0,Q_1)$ be a quiver without oriented cycle.
Here, $Q_0$ is the set of vertices and $Q_1$ the set of arrows.
Let $\beta=(\beta(s))_{s\in Q_0}$  be a vector dimension of $Q$ and
${\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$ the vector space of the representations of dimension vector $\beta$.
The group ${\rm GL}(\beta)=\prod_{s\in Q_0}{\rm GL}(\beta(s))$  acts naturally on ${\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$.
We consider the group $\Gamma$ of the characters of ${\rm GL}(\beta)$; it is isomorphic to $\ZZ^{Q_0}$.
We consider the cone $\Sigma(Q,\beta)$ in $\Gamma\otimes\QQ$ generated the elements  $\sigma\in\Gamma$
such that there exists a non zero regular function $f\in{\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\beta)]$ such that $g.f=\sigma(g)f$
for any $g\in{\rm GL}(\beta)$.
Actually, it is a convex polyhedral cone.
In \cite{DW:saturation,DW:comb}, Derksen-Weyman showed that Horn's cones can be obtained in such a way.
This is an important motivation for the study of these cones.
Here, we use general methods of Geometric Invariant Theory to give a proof of a Derksen-Weyman's Theorem (see
\cite{DW:comb}) which parametrizes bijectively the faces of $\Sigma(Q,\beta)$.\\

In Section~\ref{sec:gen}, we improve results of \cite{GITEigen} about GIT-cones in general.
In particular, Theorem~\ref{th:wc2face} is an improvement of \cite[Theorem~7]{GITEigen},
and Theorem~\ref{th:faceL} of \cite[Theorem~6]{GITEigen}.
So, the examples  $\Sigma(Q,\beta)$ enlight the general theory of GIT-cones.\\

The Horn's cones can also be obtained as GIT-cones for the action of the linear groups on products of
complete flag varieties. This point of view was used \cite{sjamaar,BK,GITEigen,GITEigen2} for example.
Whereas, in the literature this GIT approach of Horn's problem was distinct from the quiver one,
this work prove that one can use essentially the same techniques in the two cases.

\section{Well covering pairs and GIT-cones}
\label{sec:gen}

\subsection{Well covering pairs}
\label{sec:def}

Let $G$ be a reductive group acting on a smooth variety $X$.
Let $\lambda$ be a one parameter subgroup of $G$.
Let $G^\lambda$ denote the centralizer of $\lambda$ in $G$.
We consider the usual
parabolic subgroup $P(\lambda)$ associated to $\lambda$ with Levi subgroup $G^\lambda$:
$$
P(\lambda)=\left \{
g\in G \::\:
\lim_{t\to 0}\lambda(t).g.\lambda(t)^{-1}
\mbox{  exists in } G \right \}.
$$

Let $C$ be an irreducible  component of the fix point set $X^\lambda$ of $\lambda$ in $X$.
We also consider the Bialinicky-Birula cell $C^+$ associated to $C$:
$$
C^+=\{x\in X\ |\ \lim_{t\to 0}\lambda(t)x\in C\}.
$$
Then, $C$ is stable by the action of $G^\lambda$ and $C^+$ by the action of $P(\lambda)$.

Consider over $G\times C^+$ the action of $G\times P(\lambda)$ given by the formula
(with obvious notation): $(g,p).(g',y)=(gg'p^{-1},py)$.
Consider the  quotient $G\times_{P(\lambda)}C^+$ of $G\times C^+$ by the action of
$\{e\}\times P(\lambda)$.
The class of a pair $(g,y)\in G\times C^+$ in $G\times_{P(\lambda)} C^+$ is denoted by $[g:y]$.

The action of $G\times\{e\}$ induces an action of $G$ on $G\times_{P(\lambda)}C^+$.
Moreover, the first projection $G\times C^+\longrightarrow G$ induces a $G$-equivariant map
$\pi\,:\,G\times_{P(\lambda)} C^+\longrightarrow G/P(\lambda)$ which is a locally trivial fibration
with fiber $C^+$.
Consider also the $G$-equivariant map
$$\eta\,:\,G\times_{P(\lambda)}C^+\longrightarrow X,\,[g:y]\mapsto gy.$$

\begin{defin}
The pair $(C,\lambda)$ is said to be {\it dominant} if $\eta$ is.
 The pair $(C,\lambda)$ is said to be {\it well covering} if $\eta$ induces an isomorphism over
an open subset of $X$ intersecting $C$.
\end{defin}

Let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\rm Pic}^G(X)$.
Let $x$ be any point in $C$.
Since $\lambda$ fixes $x$, it induces a linear action of
the group ${\mathbb K}^*$ on the fiber ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_x$.
This action defines a character of ${\mathbb K}^*$, that is, an element of
$\ZZ$ denoted by $\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(x,\lambda)$.
One easily checks that $\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(x,\lambda)$ does not depends on $x\in C$; it will be denoted by
$\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(C,\lambda)$.

\subsection{Total cones and well covering pair}

\paragraphe
Consider the convex cones ${\mathcal{TC}}^G(X)$ generated in ${\rm Pic}^G(X)_\QQ$ by the ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$'s  in ${\rm Pic}^G(X)$
which have non zero $G$-invariant sections.
We will denote by $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ the open subset of the $x$'s in $X$ such that
some positive integer $n$, there exists a $G$-invariant section of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}^{\otimes n}$
such that $\sigma(x)\neq 0$. Note that this definition is standard, only if ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$
is ample.
Since $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})=X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}^{\otimes n})$
(for any positive integer $n$), one can define $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ if ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\rm Pic}^G(X)_\QQ$.

Let $(C,\lambda)$ be a dominant  pair. Since ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\mapsto \mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(C,\lambda)$ is a group morphism,
it induces a linear map from ${\rm Pic}^G(X)_\QQ$ to $\QQ$, also denoted by $\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(C,\lambda)$.
By \cite[Lemma~7]{GITEigen}, ${\mathcal{TC}}^G(X)$ is contained in the halfspace $\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(C,\lambda)\leq 0$.
In particular, intersecting ${\mathcal{TC}}^G(X)$ with the hyperplane $\mu^{\mathcal{L}}}\def\Mi{{\mathcal{M}}(C,\lambda)=0$, one obtains
a face ${\mathcal F}(C)$ of ${\mathcal{TC}}^G(X)$.
Indeed, the following lemma shows that the face only depends on $C$:

\begin{lemma}
\label{lem:FC}
Let $(C,\lambda)$ be a dominant pair.
Then, ${\mathcal F}(C)$ is the set of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\rm Pic}^{G}(X)_\QQ$ such that $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$
intersects $C$.
\end{lemma}

\begin{proof}
Assume that   $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$  intersects $C$.
Let $\sigma$ be a $G$-invariant section of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}^{\otimes n}$ and $x\in C$ such that
$\sigma(x)\neq 0$.
We have: $\lambda(t)\sigma(x)$ is equal on one hand to $t^{n\mu^{{\mathcal{L}}}\def\Mi{{\mathcal{M}}}(C,\lambda)}\sigma(x)$ and on the hand to
$\sigma(\lambda(t)x)=\sigma(x)$. It follows that $ \mu^{{\mathcal{L}}}\def\Mi{{\mathcal{M}}}(C,\lambda)=0$.

Conversely, let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\mathcal F}(C)$. Let $\sigma$ be a non zero $G$-invariant section of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}^{\otimes n}$.
Since $\eta$ is dominant, $\sigma$ is non identically zero on $C^+$.
By \cite[Proposition~5]{GITEigen}, $\mu^{n{\mathcal{L}}}\def\Mi{{\mathcal{M}}}(C,\lambda)=0$ implies that
 $\sigma$ is non identically zero on $C$.
In particular, $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ intersects $C$.
\end{proof}\\

\begin{remark}
  In the above proof, we have only used that $\eta$ is dominant.
\end{remark}

The following theorem is an improvement of \cite[Theorem~7]{GITEigen}.

\begin{theo}
\label{th:wc2face}
We assume the rank of  ${\rm Pic}^G(X)$ is finite and consider ${\mathcal{TC}}^G(X)$.
Let $(C,\lambda)$ be a well covering pair.

The rank of ${\rm Pic}^{G^\lambda}(C)$ is finite.
The codimension of ${\mathcal F}(C)$ in ${\rm Pic}^G(X)_\QQ$ equals the codimension of
${\mathcal{TC}}^{G^\lambda}(C)$ in ${\rm Pic}^{G^\lambda}(C)_\QQ$.
More precisely,
the restriction morphism induces an isomorphism
from ${\rm Pic}^G(X)_\QQ/\langle {\mathcal F}(C)\rangle$ onto
 ${\rm Pic}^{G^\lambda}(C)_\QQ/\langle {\mathcal{TC}}^{G^\lambda}(C)\rangle$.
\end{theo}

\begin{proof}
  Let $\Omega$ be a $G$-stable open subset of $X$ such that the natural map
$G\times_{P(\lambda)}(C^+\cap \Omega)\longrightarrow \Omega$ is an isomorphism.
Since $(C,\lambda)$ is well covering one can find such an $\Omega$ intersecting $C$.
By \cite[Lemma~1]{GITEigen}, ${\rm Pic}^G(\Omega)$ is isomorphic to ${\rm Pic}^{G^\lambda}(C\cap \Omega)$.

Let $E_1,\cdots,E_s$ (resp. $D_1,\cdots,D_t$) be the irreducible components of codimension one of
$X-\Omega$ (resp. $C-\Omega$).
Since $G$ and $G^\lambda$ are connected the $E_i$'s and the $D_i$'s are respectively $G$ and
$G^\lambda$-stable. We consider the associated $G$ and $G^\lambda$-linearized line bundles ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{E_i}$
and ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{D_i}$.

Consider the following diagram:

\begin{diagram}
\oplus_i\QQ{\mathcal{L}}}\def\Mi{{\mathcal{M}}_{E_i}&\rTo&{\rm Pic}^G(X)_\QQ \\
                    &    &             &\rdTo^{\pi_X}\\
                    &    & \dTo        &\ \ \ \ \ \  &{\rm Pic}^G(\Omega)_\QQ\simeq{\rm Pic}^{G^\lambda}(C\cap \Omega)_\QQ\\
                    &    &             &\ruTo_{\pi_C}\\
\oplus_i\QQ{\mathcal{L}}}\def\Mi{{\mathcal{M}}_{D_i}&\rTo&{\rm Pic}^{G^\lambda}(C)_\QQ
\end{diagram}

Since $X$ and so $C$ are  smooth, the maps $\pi_X$ and $\pi_C$ are surjective.
By construction, ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{D_i}$ belongs to ${\mathcal{TC}}^{G^\lambda}(C)$.
Moreover, each ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{E_i}$ has a $G$-invariant section with $E_i$ has zero locus.
Since $E_i$ does not contains $C$, this proves that ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{E_i}\in{\mathcal F}(C)$.
So, it is sufficient to prove that $\pi_X({\mathcal F}(C))=\pi_C({\mathcal{TC}}^{G^\lambda}(C))$.

Let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\mathcal F}(C)$.
Since $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ intersects $C$, ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_{|C}$ belongs to ${\mathcal{TC}}^{G^\lambda}(C)$.
So, $\pi_X({\mathcal F}(C))\subset \pi_C({\mathcal{TC}}^{G^\lambda}(C))$.

Conversely, let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$ be a $G^\lambda$-linearized line bundle on $C$ which belongs to
${\mathcal{TC}}^{G^\lambda}(C)$ and $\sigma$ be a non zero $G^\lambda$-invariant section of
${\mathcal{L}}}\def\Mi{{\mathcal{M}}$. Let $\tilde{\mathcal{L}}}\def\Mi{{\mathcal{M}}$ be the $G$-linearized line bundle on $\Omega$ associated to $\pi_C({\mathcal{L}}}\def\Mi{{\mathcal{M}})$, and
$\tilde\sigma$ the $G$-invariant section of $\tilde{\mathcal{L}}}\def\Mi{{\mathcal{M}}$ associated to $\sigma$.
Now, let $\Mi\in{\rm Pic}^G(X)$ such that $\pi_X(\Mi)=\tilde{\mathcal{L}}}\def\Mi{{\mathcal{M}}$.
The section $\tilde\sigma$ induces a non zero $G$-invariant rational section of $\Mi$, and so a
non zero regular $G$-invariant section $\sigma'$ of $\Mi'=\Mi+\sum_i {\mathcal{L}}}\def\Mi{{\mathcal{M}}_{E_i}^{\otimes n_i}$ for some non negative
integers $n_i$. Since no $E_i$ contains $C$, $\sigma'$ is not identically zero on $C$; in particular,
$\Mi'\in {\mathcal F}(C)$.
Since, $\pi_X(\Mi')=\pi_X(\Mi)=\pi_C({\mathcal{L}}}\def\Mi{{\mathcal{M}})$, it follows that $\pi_X({\mathcal F}(C))\supset \pi_C({\mathcal{TC}}^{G^\lambda}(C))$.
Note that details about the above argue can be found in the proof of \cite[Theorem~7]{GITEigen}.
\end{proof}\\

\paragraphe
The set of ample $G$-linearized line bundles generate an open convex ${\rm Pic}^G(X)^+_\QQ$ in ${\rm Pic}^G(X)_\QQ$.
We set: ${\mathcal{AC}}^G(X)={\rm Pic}^G(X)^+_\QQ\cap{\mathcal{TC}}^G(X)$.
If ${\mathcal F}$ is a face of ${\mathcal{TC}}^G(X)$, ${\mathcal F}^0$ denotes its intersection with ${\rm Pic}^G(X)^+_\QQ$.
To any ample ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in{\rm Pic}^G(X)_\QQ$ which does not belong to ${\mathcal{AC}}^G(X)$, using mainly
Kempf's Theorem, we associated in \cite{GITEigen} a well covering pair $(C,\lambda)$
(actually, defined up to conjugacy).
Note that, to do this we need to fix a ``norm'' on the set of one parameter subgroups of $G$ invariant
by conjugacy.  When restricted to $Y(S)\otimes \QQ$, (where $Y(S)$ denotes the group of one parameter
subgroups  of any subtorus $S$ of $G$), this norm becomes the norm associated to a scalar product.
The face ${\mathcal F}(C)$ is also denoted by ${\mathcal F}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$.
\cite[Theorem~6]{GITEigen} asserts that any face of ${\mathcal{AC}}^G(X)$ equals ${\mathcal F}^\circ({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ for some
ample ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\not\in{\mathcal{AC}}^G(X)$.
Here, we need an improvement of this result.

Let $K^\lambda$ denote the neutral component of the kernel of the action of $G^\lambda$ on $C$.
Note that ${\rm Pic}^{G^\lambda/K^\lambda}(C)_\QQ$ is naturally embedded in ${\rm Pic}^{G^\lambda}(C)_\QQ$;
and contains ${\mathcal{TC}}^{G^\lambda}(C)$.

\begin{theo}
\label{th:faceL}
Let ${\mathcal F}$ be a face of ${\mathcal{AC}}^G(X)$.
Consider the set $\Delta({\mathcal F})$ of ample ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\not\in{\mathcal{AC}}^G(X)$ such that ${\mathcal F}={\mathcal F}^\circ({\mathcal{L}}}\def\Mi{{\mathcal{M}})$.

Then,
\begin{enumerate}
\item There exists ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in \Delta({\mathcal F})$ such that the associated pair $(C,\lambda)$ satisfies
${\mathcal{AC}}^{G^\lambda}(C)$ has non empty interior in ${\rm Pic}^{G^\lambda/K^\lambda}(C)$.
\item $\Delta({\mathcal F})$ contains a non empty open subset of ${\rm Pic}^G(X)_\QQ^+-{\mathcal{AC}}^G(X)$.
\end{enumerate}

\end{theo}

\begin{proof}
  Let $S$ be a maximal torus of $K^\lambda$.
Let $S'$ be a torus of $G^\lambda$ such that the product induces an isogeny from $S\times S'$ onto
a maximal torus $T$
of $G^\lambda$ and $Y(S)$ is orthogonal with $Y(S')$.
Let $H^\lambda$ be the connected subgroup of $G^\lambda$ such that
the product induces an isogeny $K^\lambda\times H^\lambda\longrightarrow G^\lambda$
and containing $S'$.

Consider the restriction morphism $p\,:\,{\rm Pic}^G(X)\longrightarrow{\rm Pic}^{H^\lambda}(C)$.
Note also that ${\rm Pic}^{H^\lambda}(C)_\QQ$ is isomorphic to ${\rm Pic}^{G^\lambda/K^\lambda}(C)_\QQ$.

Let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in\Delta({\mathcal F})$.
We assume that the interior of ${\mathcal{AC}}^{G^\lambda}(C)$ in ${\rm Pic}^{G^\lambda/K^\lambda}(C)$ is empty.
By \cite[Lemma~9]{GITEigen}, $p({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ belongs to ${\mathcal{AC}}^{H^\lambda}(C)$.
By Theorem~\ref{th:wc2face}, one can find a neighbor ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_\epsilon$ of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$ such that $p({\mathcal{L}}}\def\Mi{{\mathcal{M}}_\epsilon)$
does not belong to the span of ${\mathcal{AC}}^{G^\lambda}(C)$. By \cite[Lemma~10]{GITEigen}, on can find such a
${\mathcal{L}}}\def\Mi{{\mathcal{M}}_\epsilon$ such that the face of ${\mathcal{AC}}^{H^\lambda}(C)$ viewed from ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_\epsilon$ is the whole
${\mathcal{AC}}^{H^\lambda}(C)$. By \cite[Lemma~11]{GITEigen}, one may assume that ${\mathcal F}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_\epsilon)={\mathcal F}$.
Moreover the proof of \cite[Lemma~11]{GITEigen} shows that $C_\epsilon$ (with obvious notation) is
strictly contained in $C$.
By induction on the dimension of $C$ we just proved that there exists ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in\Delta({\mathcal F})$ such that
 interior of ${\mathcal{AC}}^{G^\lambda}(C)$ in ${\rm Pic}^{G^\lambda/K^\lambda}(C)$ is non empty.\\

By the same argue as above, one can prove that there exists ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in\Delta({\mathcal F})$
such that $p({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ belongs to the interior of
${\mathcal{AC}}^{H^\lambda}(C)\cap {\rm Im}p$ in ${\rm Im}p$.
In this case, $p^{-1}({\mathcal{AC}}^{H^\lambda}(C))\cap ({\rm Pic}^G(X)^+_\QQ-{\mathcal{AC}}^G(X))$ has non empty interior in
${\rm Pic}^G(X)_\QQ$. Moreover, by \cite[Lemma~10]{GITEigen}, this set is contained in $\Delta({\mathcal F})$.
The second assertion is proved.
\end{proof}

\section{Application to quiver representations}

\subsection{Definitions}
\label{sec:defcarquois}

In this section, we fix some classical notation about quiver representations.

Let $Q$ be a quiver (that is, a finite oriented graph) with vertexes $Q_0$ and arrows $Q_1$.
An arrow $a\in Q_1$ has initial vertex $ia$ and terminal one $ta$.
A representation $R$ of $Q$ is a family $(V(s))_{s\in Q_0}$ of finite dimensional vector spaces and
a family of linear maps $u(a)\in {\rm Hom}(V(ia),V(ta))$ indexed by $a\in Q_1$.
The dimension vector of $R$ is the family $(\dim(V(s)))_{s\in Q_0}\in \NN^{Q_0}$.

Let us fix $\alpha\in \NN^{Q_0}$ and  a vector space $V(s)$ of dimension $\alpha(s)$ for each $s\in Q_0$.
Set
$$
{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)=\bigoplus_{a\in Q_1}{\rm Hom}(V(ia),V(ta)).
$$
Consider also the groups:
$$
{\rm GL}(\alpha)=\prod_{s\in Q_0}{\rm GL}(V(s))
{\rm\ and\ }
\SL(\alpha)=\prod_{s\in Q_0}\SL(V(s).
$$
They acts naturally on ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$.

The character group of ${\rm GL}(\alpha)$ identifies with $\Gamma=\ZZ^{Q_0}$; to
$\sigma\in \ZZ^{Q_0}$, we associate the character $\chi_\sigma$ defined by
 $\chi_\sigma\left( (g(s))_{s\in Q_0})\right)=\prod_{s\in Q_0}\det(g(s))^{\sigma(s)}$.

\subsection{Three cones}

\paragraphe
Consider the algebra ${\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]$ of regular functions on ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$ endowed with the ${\rm GL}(\alpha)$-action.
For $\sigma\in \ZZ^{Q_0}$, we denote ${\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]_\sigma$ the set of $f\in{\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]$ such that
for all $g\in{\rm GL}(\alpha)$, $g.f=\chi_\sigma(g)f$.
We embed $\Gamma=\ZZ^{Q_0}$ in $\Gamma_\QQ:=\QQ^{Q_0}$.
Let $\Sigma(Q,\alpha)$ denote the convex cone of $\Gamma_\QQ$ generated by the points
$\sigma\in\Gamma$ such that ${\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]_{-\sigma}$ is non reduced to $\{0\}$.

\paragraphe
Consider the projective space $X={\mathbb P}}\def\NN{{\mathbb N}({\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\oplus {\mathbb C})$. The formula
$$
g.(R,t)=(gR,t)\ \ \ \forall g\in{\rm GL}(\alpha)\,,R\in{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha){\rm\ and\ }
t\in{\mathbb C},
$$
defines an action of ${\rm GL}(\alpha)$ on $X$ and a ${\rm GL}(\alpha)$-linearization ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_0\in{\rm Pic}^{{\rm GL}(\alpha)}(X)$ of the line
bundle ${\mathcal O}(1)$ on $X$.
We are now interested in the GIT-cone ${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)$.
Since any line bundle on $X$ admitting non zero sections is ample, ${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)={\mathcal{TC}}^{{\rm GL}(\alpha)}(X)$.

For $n\in \ZZ$ and $\sigma\in \Gamma$, set ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(n,\sigma)={\mathcal{L}}}\def\Mi{{\mathcal{M}}_0\otimes\sigma\in{\rm Pic}^{{\rm GL}(\alpha)}(X)$.
Note that ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(n,\sigma)={\mathcal O}(n)$ as a   line bundle.
We have the following obvious

\begin{lemma}
\label{lem:PicGX}
  The map $\ZZ\times\Gamma\longrightarrow{\rm Pic}^{{\rm GL}(\alpha)}(X),\,(n,\sigma)\mapsto{\mathcal{L}}}\def\Mi{{\mathcal{M}}(n,\sigma)$ is an isomorphism of groups.
Moreover, ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(n,\sigma)$ is ample if and only if $n$ is positive.
\end{lemma}

Lemma~\ref{lem:PicGX} allows to embed ${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)$ in $\QQ\times\Gamma_\QQ$.
Set ${\mathcal{P}}(Q,\alpha)={\mathcal{AC}}^{{\rm GL}(\alpha)}(X)\cap \{1\}\times\Gamma_\QQ$.
General properties of ${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)$ imply that ${\mathcal{P}}(Q,\alpha)$ is  closed convex rational and polyhedral.
Moreover, it is contained in the convex hull of the restrictions to the center of ${\rm GL}(\alpha)$ of the weights
of a maximal torus of ${\rm GL}(\alpha)$ on ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\oplus{\mathbb C}$; and so, it is compact.
Finally,  ${\mathcal{P}}(Q,\alpha)$ is a rational polytope in $\Gamma_\QQ$.
Moreover, ${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)$ is the convex cone generated by ${\mathcal{P}}(Q,\alpha)$ in such a  way the faces of
${\mathcal{AC}}^{{\rm GL}(\alpha)}(X)$ and ${\mathcal{P}}(Q,\alpha)$ correspond bijectively.
\\

\paragraphe
We consider ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$ as an open subset of $X$ by $R\mapsto (R,1)$; and we identify the complement
with ${\mathbb P}}\def\NN{{\mathbb N}({\rm Rep}}\def\SI{{\rm SI}(Q,\alpha))$.

\begin{prop}
\label{prop:0sommet}
\begin{enumerate}
\item We have: $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_0)={\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$.
\item The point $0\in\Gamma_\QQ$ is a vertex of ${\mathcal{P}}(Q,\alpha)$.
\item The cone of $\Gamma_\QQ$ generated by ${\mathcal{P}}(Q,\alpha)$ is $\Sigma(Q,\alpha)$.
\end{enumerate}
\end{prop}

\begin{proof}
Since $Q$ has no oriented cycle, one can chose a numeration of the vertices such that
the index of $ta$ is greater than the index of $ia$ for all $a\in Q_1$.
Consider the one parameter subgroup $\lambda_0$ of ${\rm GL}(\alpha)$ acting on the vector space
corresponding to the vertex indexed by $i$ as an homothetie of coefficient $t^i$.

The point $0\in{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\subset X$ is an isolated fixed point of $\lambda_0$.
Set $C_0=\{0\}$. One easily checks that $C_0^+={\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$ and that $\lambda_0$ is central in
${\rm GL}(\alpha)$: it follows that $(C_0,\lambda_0)$ is a well covering pair.
Let ${\mathcal F}(C_0)$ (resp. ${\mathcal{P}}(C_0)$) denote the face of
${\mathcal{TC}}^{{\rm GL}(\alpha)}(X)$ (resp. ${\mathcal{P}}(Q,\alpha)$) associated to $(C_0,\lambda_0)$.

Viewed the action of the center of ${\rm GL}(\alpha)$ on the fiber over $0$,
 ${\mathcal F}(C_0)$ is contained in $\QQ^+{\mathcal{L}}}\def\Mi{{\mathcal{M}}_0$.

  Since $(R,t)\mapsto t$ is a ${\rm GL}(\alpha)$-invariant section of ${\mathcal{L}}}\def\Mi{{\mathcal{M}}_0$,
${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\subset X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_0)$.
Then, ${\mathcal F}(C_0)=\QQ^+{\mathcal{L}}}\def\Mi{{\mathcal{M}}_0$.

Since $0$ is the only closed ${\rm GL}(\alpha)$-orbit in ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$,
$X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_0)/\hspace{-.5ex}/ {\rm GL}(\alpha)$ is only one point.
So, $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_0)$ contains only one closed orbit ${\mathcal O}$ which is contained in the closure
${\rm GL}(\alpha).0$. We deduce that ${\mathcal O}=\{0\}$ and that $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}_0)={\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$.

The last assertion of the proposition is a direct application of \cite[Theorem~4]{GITEigen2}.
\end{proof}

\paragraphe
Consider now the projective space $D={\mathbb P}}\def\NN{{\mathbb N}({\rm Rep}}\def\SI{{\rm SI}(Q,\alpha))$ endowed with the
${\rm GL}(\alpha)$-action.
We are now interested in the GIT-cone ${\mathcal{AC}}^{{\rm GL}(\alpha)}(D)$.

We have the following obvious

\begin{lemma}
\label{lem:PicGD}
  The restriction map $\rho_D\,:\,{\rm Pic}^{{\rm GL}(\alpha)}(X)\longrightarrow{\rm Pic}^{{\rm GL}(\alpha)}(D)$ is an isomorphism of groups.
Moreover, $\rho_D({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ is ample if and only if ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$ is.
\end{lemma}

Lemma~\ref{lem:PicGD} allows to embed ${\mathcal{AC}}^{{\rm GL}(\alpha)}(D)$ in $\QQ\times\Gamma_\QQ$.
Set ${\mathcal{P}}(D,Q,\alpha)={\mathcal{AC}}^{{\rm GL}(\alpha)}(D)\cap \{1\}\times\Gamma_\QQ$.
Obviously,   ${\mathcal{P}}(D,Q,\alpha)$ is a rational polytope in $\Gamma_\QQ$.
\\

Via the identification of Lemma~\ref{lem:PicGD}, the relation between
${\mathcal{P}}(D,Q,\alpha)$ and ${\mathcal{P}}(Q,\alpha)$ is as follows:

\begin{prop}
\label{prop:PDalpha}
The polytope ${\mathcal{P}}(Q,\alpha)$ is the convex hull of $0$ and
${\mathcal{P}}(D,Q,\alpha)$.

We assume moreover that $Q$ has no cycle (even non oriented). Then,
${\mathcal{P}}(D,Q,\alpha)$ is a face of ${\mathcal{P}}(Q,\alpha)$.
In particular, ${\mathcal{P}}(D,Q,\alpha)$ is an affine section of $\Sigma(Q,\alpha)$.
\end{prop}

\begin{proof}
Let $\sigma\in\Gamma_\QQ$.
It is clear that $\sigma\in{\mathcal{P}}(D,Q,\alpha)$  if and only if
$X^{\rm ss}(\sigma)$ intersects $D$,  if and only if
$X^{\rm ss}(\sigma)$ is not contained in $X^{\rm ss}(0)$.
By \cite{GeomDedic}, this is equivalent to the fact that
the closure of the GIT-class of $\sigma$ does not contain $0$.
In particular, all the vertices of ${\mathcal{P}}(Q,\alpha)$ excepted $0$ belong
to ${\mathcal{P}}(D,Q,\alpha)$; the first assertion follows.

With the additional assumption, one can easily construct a central one
parameter subgroup $\lambda(t)$ of $G(\alpha)$ which acts on
each ${\rm Hom}(V(ia),V(ta))$ (for $a\in Q_1$) by multiplication by $t$.
Then, $(D,\lambda)$ is a well covering pair; this implies that
${\mathcal{P}}(D,Q,\alpha)$ is a face of ${\mathcal{P}}(Q,\alpha)$.
\end{proof}\\

\paragraphe
Propositions~\ref{prop:0sommet} and \ref{prop:PDalpha} proves that when $Q$ has no cycle,
the descriptions of the three cones are equivalent. From now on, we are mainly
interested in $\Sigma(Q,\alpha)$ viewed as the cone generated by ${\mathcal{P}}(Q,\alpha)$;
that is to the faces of ${\mathcal{P}}(Q,\alpha)$ containing the vertex $0$:

\begin{lemma}
\label{lem:C0}
Let $(C,\lambda)$ be a dominant pair.
Then, ${\mathcal F}(C)$ contains $0$ if and only if $C$ contains $0$.
\end{lemma}

\begin{proof}
  The point is that $\{0\}$ is the only closed orbit in $X^{\rm ss}(0)$.
Actually, if ${\mathcal F}(C)$ contains $0$, $C$ has to contains $0$ by
\cite[Theorem~5]{GITEigen}. Conversely, if $C$ contains $0$, $\mu^0(C,\lambda)=0$.
\end{proof}

\subsection{Dominant pairs}

\paragraphe\label{par:lambdaalpha}
Let $\sigma\in\Gamma$ and $\alpha$ a vector dimension. We set:
$$
\sigma(\alpha):=\sum_{s\in Q_0}\sigma(s)\alpha(s).
$$
We consider the one parameter subgroup $\lambda_\alpha$ of ${\rm GL}(\alpha)$ acting on $V(s)$ by $t.{\rm Id}}\def\Ker{{\rm Ker}$ for any
$s\in Q_0$: $\sigma(\alpha)$ is simply the composition $\sigma\circ\lambda_\alpha$.
Note that $\lambda_\alpha$ acts trivially on ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$.
This implies that ${\mathcal{P}}(Q,\alpha)$ is contained in the hyperplane ${\mathcal H}}\def\Wcal{{\mathcal W}(\alpha)$ consisting of the $\sigma$'s
such that $\sigma(\alpha)=0$.\\

\begin{defin}
  The dimension vector $\alpha$ is called a {\it rational Schur root} if ${\mathcal{P}}(Q,\alpha)$ or equivalently $\Sigma(Q,\alpha)$
has non empty interior in ${\mathcal H}}\def\Wcal{{\mathcal W}(\alpha)$.

If there exists  $R\in{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$ whose the stabilizer in ${\rm GL}(\alpha)$ has dimension one, $\alpha$ is said to be
a {\it Schur root}.
\end{defin}

The second notion is very classical (see \cite{Kac:carquois2}) and the first one very natural in our context.
We will explain the relation between these two notions in Paragraph~\ref{par:ratSchur}.\\

\paragraphe {\bf Decompositions of dimension vectors.}
Let $\alpha$ be a vector dimension of $Q$.\\

\begin{defin}
A {\it $\ZZ$-decomposition} of $\alpha$ is a family of dimension vectors $\alpha_i$ indexed by $\ZZ$
such that $\alpha_i=0$ with finitely many exceptions and $\alpha=\sum_{i\in\ZZ}\alpha_i$.
An {\it ordered decomposition} of $\alpha$, is a sequence $(\beta_1,\cdots,\beta_s)$ of
non-zero vector dimensions such that $\alpha=\beta_1+\cdots+\beta_s$.
We denote the decomposition by $\alpha=\beta_1\pplus\cdots\pplus\beta_s$.
\end{defin}

\paragraphe \label{par:defeta}
Let $\lambda$ be a one parameter subgroups of ${\rm GL}(\alpha)$.
For any $i\in \ZZ$ and $s\in Q_0$, we set $V_i(s)=\{v\in V(s)\,|\,\lambda(t)v=t^iv\}$ and
$\alpha_i(s)=\dim V_i(s)$.
Obviously, $\alpha=\sum_{i\in\ZZ}\alpha_i$ form a $\ZZ$-decomposition of $\alpha$ which
determines $\lambda$ up to conjugacy.

The parabolic subgroup $P(\lambda)$ of ${\rm GL}(\alpha)$ associated to $\lambda$ is the set of
$(g(s))_{s\in Q_0}$ such that for all $i\in\ZZ$ and $s\in Q_0$ we have
$g(s)(V_i(s))\subset \oplus_{j\leq i}V_j(s)$.

Now, ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)^\lambda$ is the set of the $(u(a))_{a\in Q_1}$'s such that
for any $a\in Q_1$ and for any $i\in\ZZ$, $u(a)(V_i(ia))\subset V_i(ta)$.
It is  isomorphic to $\prod_i{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha_i)$.
In particular, the irreducible component $C$ of $X^\lambda$ containing $0$ is isomorphic
to ${\mathbb P}}\def\NN{{\mathbb N}(\prod_i{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha_i)\oplus{\mathbb C})$.

Moreover, $C^+\cap {\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$ is the set of the $(u(a))_{a\in Q_1}$'s such that
for any $a\in Q_1$ and for any $i\in\ZZ$, $u(a)(V_i(ia))\subset \oplus_{j\leq i}V_j(ta)$.

Consider the morphism $\eta_\lambda\,:\,G\times_{P(\lambda)}C^+\longrightarrow {\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)$.
Note that, $P(\lambda)$, $C$ and $C^+$ only depend (up to conjugacy) on the ordered decomposition of
$\alpha$ induced by the $\ZZ$-decomposition $\sum_i\alpha_i$ in an obvious way. From now on, we will
consider the
{\it map $\eta_{\beta_1\pplus\cdots\pplus\beta_s}$} associated to the ordered decomposition of $\alpha$;
it is defined up to conjugacy.
We will say that the ordered decomposition is dominant respectively birational if
$\eta_{\beta_1\pplus\cdots\pplus\beta_s}$ is. We will say that the decomposition is {\it well covering}
if $(C,\lambda)$ is.
If the decomposition is dominant, we will denote by
${\mathcal F}(\beta_1\pplus\cdots\pplus\beta_s)$ the corresponding face of ${\mathcal{P}}(Q,\alpha)$.

\begin{lemma}
\label{lem:descFdecomp}
Let $\beta=\beta_1\pplus\cdots\pplus\beta_s$ be a dominant ordered decomposition.
Then, ${\mathcal F}(\beta_1\pplus\cdots\pplus\beta_s)$ is generated by the face
${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)\cap\Sigma(Q,\beta)$ of $\Sigma(Q,\beta)$.
\end{lemma}

\begin{proof}
  Let $(C,\lambda)$ be a dominant pair associated to  $\beta=\beta_1\pplus\cdots\pplus\beta_s$.
Let us fix $\underline V=(V(s))_{s\in Q_0}$ of dimension $\beta$.
Let $\underline V=\underline V_1\oplus\underline V_s$ be a decomposition such that $\underline V_i$
has dimension $\beta_i$.
The torus $({\mathbb C}^*)^s$ acts on $\underline V$ as follows; the $i^{\rm th}$ component acts by homothetie
on $\underline V_i$.
The induced action of $({\mathbb C}^*)^s$ on $C$ is trivial.
So, $({\mathbb C}^*)^s$ has to acts trivially on any point in ${\mathcal F}(C)$; it follows that,
${\mathcal F}(C)$ is contained in ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)$.

Conversely, let $\sigma\in {\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)\cap\Sigma(Q,\beta)$.
Since $(C,\lambda)$ is covering, $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma))$ intersects $C^+$.
But, since $\sigma\in {\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)$, $\lambda$ acts trivially
on ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma)_{|C}$. By \cite[Lemma~4]{GITEigen}, this implies that $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma))$
intersects $C$.
\end{proof}\\

\paragraphe
Let $\alpha,\beta\in\NN^{Q_0}$.
Following Derksen-Schofield-Weyman (see~\cite{DSW:nbsubrep}), we define $\alpha\circ\beta$ to be the
number of $\alpha$-dimensional subrepresentation of a general representation of dimension $\alpha+\beta$
if it is finite, and $0$ otherwise.

We now recall a description of well covering ordered decomposition from \cite{multi}:
\begin{prop}
\label{prop:wcod}
The ordered decomposition $\beta= \beta_1\pplus\cdots\pplus\beta_s$ is well covering if and only if
$$\forall i<j\ \ \beta_i\circ \beta_j=1.
$$
\end{prop}

\paragraphe
\label{par:ratSchur}
We can now explain the name ``rational Schur root''.
Let us first reprove two well known lemmas:

\begin{lemma}
\label{lem:alphabetagroup}
If $\alpha\circ\beta\neq 0$ and $\alpha\circ\gamma\neq 0$ then
$\alpha\circ(\beta+\gamma)\neq 0$.
\end{lemma}

\begin{proof}
In \cite{DSW:nbsubrep}, Derksen-Schofield-Weyman proved that $\alpha\circ\beta$ is the dimension of
${\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]_\sigma$ for well chosen weight $\sigma$. With this characterization, the lemma
just follows from the fact that ${\mathbb C}[{\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)]^{\SL(\alpha)}$ is an algebra.
In this work, $\alpha\circ\beta$ is always the degree a map $\eta$; and, we include a proof using this
point of view.

Consider a pair $(C,\lambda)$ (resp. $(C',\lambda'))$ associated to the ordered decomposition
$\alpha\pplus\beta$ (resp. $\alpha\circ\gamma$) in ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha+\beta)$ and ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha+\gamma)$.
Since $\alpha\circ\beta\neq 0$, $\eta_{\alpha\pplus\beta}$ is generically finite.
Moreover, by \cite[Lemma~]{multi}, $\lambda$ acts trivially on the restriction to $C$ of the determinant
bundle of $\eta$. It follows that for general $x\in C={\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\oplus{\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$, the differential of
$\eta_{\alpha\pplus\beta}$ at $x$ is an isomorphism.
In the same way, the differential of $\eta_{\alpha\pplus\gamma}$ is an isomorphism for $x'$ general in
${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\oplus{\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$.
A direct computation implies now that $\eta_{\alpha\pplus(\beta+\gamma)}$ is an isomorphism
for $y$ general in  ${\rm Rep}}\def\SI{{\rm SI}(Q,\alpha)\oplus{\rm Rep}}\def\SI{{\rm SI}(Q,\beta)\oplus{\rm Rep}}\def\SI{{\rm SI}(Q,\gamma)\subset{\rm Rep}}\def\SI{{\rm SI}(\alpha+\beta+\gamma)$.
In particular, $\alpha\circ(\beta+\gamma)\neq 0$.
\end{proof}

\begin{lemma}
  \label{lem:Sigmafirstdesc}
We have:
$$
\Sigma(Q,\beta)=\{\sigma\in\Gamma\,:\,\sigma(\beta)=0{\rm\ and\ }\sigma(\alpha)\leq 0\ \forall\alpha
{\rm\ s.t.\ }\alpha\circ(\beta-\alpha)\neq 0\}.
$$
\end{lemma}

\begin{proof}
Let $\sigma\in\Sigma(Q,\beta)$.
We already saw that $\sigma(\beta)=0$.
Let $\alpha$ be such that $\alpha\circ(\beta-\alpha)\neq 0$.
Since $\eta_{\alpha\pplus(\beta-\alpha)}$ is dominant, $\sigma(\alpha)\leq 0$.

The converse inclusion is a direct consequence of \cite{King:moduli}
(see~\cite[Remark~5]{DW:saturation}).
\end{proof}

Here, comes an easy variant of Derksen-Schofield-Weyman's saturation Theorem:

\begin{lemma}
  \label{lem:Sigmakbeta}
We have:
$$
\Sigma(Q,k\beta)=\Sigma(Q,\beta).
$$
\end{lemma}

\begin{proof}
The inclusion   $\Sigma(Q,k\beta)\subset \Sigma(Q,\beta)$ is a direct consequence of
Lemmas~\ref{lem:alphabetagroup} and \ref{lem:Sigmafirstdesc}.

The converse inclusion follows from Derksen-Weyman's Reciprocity Property
(see \cite[Corollary~1]{DW:saturation}). We include here a simpler proof.
Let $(V(s))_{s\in Q_0}$ be vector spaces of dimension vector $\beta$.
Consider $({\rm Hom}({\mathbb C}^k,V(s))_{s\in Q_0}$ as a $Q_0$ family of vector spaces of dimension vector $k\beta$.
Then, for the natural inclusion  ${\rm Rep}}\def\SI{{\rm SI}(Q,\beta)\subset {\rm Rep}}\def\SI{{\rm SI}(Q,k\beta)$,  ${\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$ is the fix point set of
$H=({\rm GL}_k)^{Q_0}\subset{\rm GL}(k\beta)$. Moreover, the centralizer of $({\rm GL}_k)^{Q_0}$ in ${\rm GL}(k\beta)$ is isomorphic
to ${\rm GL}(\beta)$. By a Luna's Theorem (see \cite{Luna:adh}), for any linearized ample line bundle a point
$x\in {\rm Rep}}\def\SI{{\rm SI}(Q,\beta)$ is semistable for ${\mathcal{L}}}\def\Mi{{\mathcal{M}}$ and the action of ${\rm GL}(\beta)$ if and only if it is for the
action of ${\rm GL}(k\beta)$.
It follows that ${\mathcal{P}}(Q,\beta)\subset{\mathcal{P}}(Q,k\beta)$.
The lemma is proved.
\end{proof}

\begin{prop}
  \label{prop:}
A vector dimension $\alpha$ is a rational Schur root if and only if it is positively proportional
to a Schur root.
\end{prop}

\begin{proof}
The Ringle form is denoted by  $\langle\cdot,\cdot\rangle$.
Let $\beta$ be a Schur root.
By \cite[Theorem~6.1]{Scho:genrep}, $X$ contains stable points for the action of
${\rm GL}(\beta)/{\rm Im}(\lambda_0)$
and the line bundle ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\langle\beta,\cdot\rangle-\langle\cdot,\beta\rangle)$.
It follows that $\Sigma(Q,\beta)$ has non empty interior in ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta)$.
By Lemma~\ref{lem:Sigmakbeta}, $k\beta$ is a rational Schur root for any positive integer $k$.

Conversely, let $\beta$ be a rational Schur root. Let $d$ denote the gcd of the $\beta(s)$ for $s\in Q_0$.
By Lemma~\ref{lem:Sigmakbeta}, $\overline{\beta}=\beta/d$ is a rational Schur root.
Consider the canonical decomposition $\overline{\beta}=\beta_1+\cdots+\beta_s$ of $\beta$
(see \cite{Kac:carquois2}).
Then, $\Sigma(Q,\overline{\beta})$ is contained in ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)$.
Since  $\Sigma(Q,\overline{\beta})$ spans the hyperplane ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta)$, it follows that
${\mathcal H}}\def\Wcal{{\mathcal W}(\beta)={\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)=\cdots={\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)$.
So, the $\beta_i$'s are proportional; since, $\overline{\beta}$ is indivisible, it follows that $s=1$ and
that $\overline{\beta}$ is a Schur root.
\end{proof}

\subsection{Derksen-Weyman's Theorem}

\paragraphe
The ordered decomposition $\beta= \beta_1\pplus\cdots\pplus\beta_s$ is called an
{\it ordered decomposition by rational Schur roots} if $\beta_1,\cdots,\beta_s$ are rational
Schur roots.
To any such decomposition we associate the (unordered) set $\{\beta_1,\cdots,\beta_s\}\subset \NN^{Q_0}$.
Let $\Wcal_s(\beta)$ denote the set of subsets  obtained in such a way from
well covering ordered decomposition by $s$ rational Schur roots.

We can now state and prove Derksen-Weyman's Theorem:

\begin{theo}
\label{th:main}
Let $\beta$ be a vector dimension. Let $d$ denote the dimension of $\Sigma(Q,\beta)$ and $n$ the cardinality of
$Q_0$.
For any $s=n-d,\cdots,0$, the map
$$
\begin{array}{cccc}
  \Theta\ :&\Wcal_s(\beta)&\longrightarrow&\{{\rm faces\ of\ }\Sigma(Q,\beta){\rm\ of\ codimension\ }d\}\\
&\{\beta_1,\cdots,\beta_s\}&\longmapsto&{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)\cap\Sigma(Q,\beta),
\end{array}
$$
is a bijection.
Moreover, the family $(\beta_1,\cdots,\beta_s)$ is linearly independent.
\end{theo}

\begin{proof}
In this proof, we prefer to consider the faces of ${\mathcal{P}}(Q,\alpha)$ containing $0$ rather than faces of
$\Sigma(Q,\alpha)$. By Proposition~\ref{prop:0sommet}, this is equivalent.

Let $\beta= \beta_1\pplus\cdots\pplus\beta_s$ be a well covering ordered decomposition by rational Schur roots.
Then,  by Lemma~\ref{lem:descFdecomp}, ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)\cap\Sigma(Q,\beta)$ is the face
of $\Sigma(Q,\beta)$ corresponding to ${\mathcal F}(\beta_1\pplus\cdots\pplus\beta_s)$.
Since the $\beta_i$'s are rational Schur roots, Theorem~\ref{th:wc2face} shows that
${\mathcal F}(\beta_1\pplus\cdots\pplus\beta_s)$ has codimension $s$.
Let us recall that ${\mathcal{AC}}^{{\rm GL}(\beta)}(X)={\mathcal{TC}}^{{\rm GL}(\beta)}(X)$.
This proves that $\Theta$ is well defined.\\

Let us fix a face ${\mathcal F}$ of ${\mathcal{P}}(Q,\alpha)$ of codimension $d$.
By Theorem~\ref{th:faceL}, the exists an open subset $U$ in ${\rm Pic}^G(X)^+_\QQ-{\mathcal{AC}}^G(X)$ such that ${\mathcal F}={\mathcal F}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$
for all ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in U$.
Let $(C,\lambda)$ be a well covering pair associated to a line bundle ${\mathcal{L}}}\def\Mi{{\mathcal{M}}\in U$: by Lemma~\ref{lem:C0}, $C$ contains $0$.
Let $\beta=\beta_1\pplus\cdots\pplus\beta_s$ be the ordered decomposition associated to $\lambda$.
By Paragraph~\ref{par:defeta}, $\eta_{(C,\lambda)}=\eta_{\beta_1\pplus\cdots\pplus\beta_s}$.

We claim that the $\beta_i$'s are rational Schur roots.
Let us fix $i\in\{1,\cdots,s\}$. Let $\lambda_{\beta_i}$ be the central one parameter subgroup
of ${\rm GL}(\beta_i)$ defined in Paragraph~\ref{par:lambdaalpha}; and, $S_i$ be the codimension one subtorus of
the center of  ${\rm GL}(\beta_i)$ such that $Y(S_i)$ is orthogonal to $\lambda_{\beta_i}$.
Consider the subgroup $H_i$ of ${\rm GL}(\beta_i)$ generated by the $S_i$ and $\SL(\beta_i)$.
We embed ${\mathbb P}}\def\NN{{\mathbb N}({\rm Rep}}\def\SI{{\rm SI}(Q,\beta_i)\oplus {\mathbb C})$ in $X$ in an obvious way and consider the restriction morphism:
$$
p_i\,:\,{\rm Pic}^G(X)_\QQ\longrightarrow{\rm Pic}^{H_i}({\mathbb P}}\def\NN{{\mathbb N}({\rm Rep}}\def\SI{{\rm SI}(Q,\beta_i)\oplus {\mathbb C}))_\QQ.
$$
By construction, the restriction of $p_i$ to ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_i)$ is surjective.
Moreover, by \cite[Lemma~11]{GITEigen}, $p_i(U)$ is contained in ${\mathcal{P}}(Q,\beta_i,H_i)$.
Since $p_i$ is an open map, this implies that ${\mathcal{P}}(Q,\beta_i)$ has codimension one in $X({\rm GL}(\beta_i))_\QQ$.
So, the $\beta_i$'s are rational Schur roots and $\Theta$ is surjective. \\

Let $\beta= \beta_1\pplus\cdots\pplus\beta_s$ be any well covering ordered decomposition by rational
Schur roots.
By Theorem~\ref{th:wc2face}, the intersection ${\mathcal H}}\def\Wcal{{\mathcal W}(\beta_1)\cap\cdots\cap{\mathcal H}}\def\Wcal{{\mathcal W}(\beta_s)$ has codimension $s$.
This means that the $\beta_i$ are linearly independent.

Let us fix $\sigma\in\Gamma_\QQ$ such that ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma)$ belongs to the relative interior of
${\mathcal F}:={\mathcal F}( \beta_1\pplus\cdots\pplus\beta_s)$.
Since, $\beta_i$ are rational Schur roots, Theorem~\ref{th:wc2face} shows that the codimension of ${\mathcal F}$
equals $s$.
Note that, $\Delta({\mathcal F})$ contains ${\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma)$ in its closure.
We claim that $p_i(\sigma)$ belongs to the relative interior of $\Sigma(Q,\beta_i)$.
Actually, if not, one can find $\sigma_\epsilon$ in $\Delta({\mathcal F})$ such that
$p_i(\sigma_\epsilon)$ does not belongs to  $\Sigma(Q,\beta_i)$.
Then, the ordered decomposition associated to $\sigma_\epsilon$ contains strictly more than $s$ vector
dimensions. By Theorem~\ref{th:wc2face} this implies that the codimension of ${\mathcal F}$ is strictly greater
than $s$; which is a contradiction.\\

We now want to prove the injectivity of $\Theta$.
Let $\beta= \beta_1\pplus\cdots\pplus\beta_s$ be a well covering ordered decomposition by rational
Schur roots and ${\mathcal F}$ be the associated face.
We want to obtain the decomposition of $\beta$ from ${\mathcal F}$.
By Proposition~\ref{prop:} and \cite[Theorem~3.2]{Scho:genrep}, the canonical decomposition of
$\beta_i=a_i\overline{\beta}_i$ for some positive integer $a_i$ and some Schur root $\overline{\beta_i}$.
Set $C={\mathbb P}}\def\NN{{\mathbb N}(\oplus_i{\rm Rep}}\def\SI{{\rm SI}(Q,\beta_i)\oplus{\mathbb C})$ and
$C_0={\mathbb P}}\def\NN{{\mathbb N}(\oplus_i{\rm Rep}}\def\SI{{\rm SI}(Q,\overline{\beta}_i)^{\oplus a_i}\oplus{\mathbb C})$; and fix embeddings $C_0\subset C\subset X$.

Let ${\mathcal{L}}}\def\Mi{{\mathcal{M}}:={\mathcal{L}}}\def\Mi{{\mathcal{M}}(1,\sigma)$ be a point in the relative interior of ${\mathcal F}$.
Let $x$ be a general point in $C_0$.
Since $p_i(\sigma)$ belongs to the relative interior of $\Sigma(Q,\overline{\beta}_i)$,
\cite[Theorem~6.1]{Scho:genrep} implies that the orbit of $x$ by the group
$\prod_i{\rm GL}(\overline{\beta}_i)^{a_i}$ is closed in $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$.
By \cite{Luna:adh}, this implies that ${\rm GL}(\beta).x$ is closed in  $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$.
Conversely, by \cite[Theorem~5]{GITEigen}, any general closed closed orbit in $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ intersects
$C$ and so $C_0$.
This proves that a general closed isotropy in $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ contains a general point of $C_0$.
In particular, any point in a general closed isotropy of $X^{\rm ss}({\mathcal{L}}}\def\Mi{{\mathcal{M}})$ decompose as
a sum of $a_1$ indecomposable representations of dimension $\overline{\beta}_1$\dots and $a_s$
 indecomposable representations of dimension $\overline{\beta}_s$.
Moreover, such a decomposition is unique and the $\overline{\beta}_i$'s are pairwise distinct
(the family is free).
The injectivity follows.
\end{proof}

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


In the far north east of Scotland, located between the Cairngorm mountains and the coast, lies the granite city of Aberdeen. This architecturally unique city may not be a typical holiday destination but campervan hire in Aberdeen has so much to offer as a holiday option! Its northern position means that in summer it is light until 11 pm and there is even the chance to see the Northern Lights. Whether you are looking for an adventure in the great outdoors, fascinating history or a relaxing break motorhome hire from a private owner with Goboony in Aberdeen gives you the perfect opportunity to explore this fantastic corner of the country.
Great new VW bus, clean and comfy for driving around. Not really suitable for older and less nimble occupants. Need to limit your luggage especially if more than 2 adults are travelling. Alan is a great host, to be recommended. We had a great trip around Scotland.
There are a variety of options for campsites near Aberdeen from coastal views to idyllic forest locations, you simply need to narrow down what is essential for your motorhome hire holiday. Within 20 minutes of the Cairngorms National Park, Feughside Caravan Park is the perfect location to escape the modern world and be immersed in nature. There is easy access to the area's many forest trails and even a fishing lake a short walk away! Park up your motorhome or campervan rental and discover Aberdeenshire's wonderful natural assets. If you are looking for a more coastal location head for Stonehaven Queen Elizabeth Park Caravan Club Site just down the road from Aberdeen for fabulous views over the sea. There is even an art-deco Olympic sized pool a short walk from the campsite if you want to get in your laps during your stay!
You certainly not be given a chance to get bored in Aberdeen with a huge variety of things to do in this bustling student city. Aberdeen's important strategic location on the coast at the mouths of the rivers Dee and Don, has provided it with wealth and political significance and as a consequence numerous fascinating historical attractions. Visitors to the city will not want to miss it's fortified cathedral in Old Aberdeen. St. Machars is a striking 15th century monument full of intrigue and the purported resting place of the arm of Williams Wallace! The quaint Footdee or 'Fittee' in Doric the local ancient dialect, is a former 19th century fishing village also not to be missed with it's charming fisherman's cottages and cobbled lanes. You can also still get some delicious seafood here! Nature lovers will be delighted with the opportunity to see a variety of species of dolphins and even whales on a short cruise from the city harbour. Aberdeen also has some great music venues with the Lemon Tree being a firm favourite with its excellent beer selection and top musical talent.
Whatever your interests you are sure to find the perfect event for you in Aberdeen! The Doric culture of the region, with its particular music and dance is something that you have to experience during your trip in the region. Open air settings give these cultural events an unforgettable character! Foodies will love the rich variety of local produce in this region with frequent fairs and festivals celebrating the region's delicious seafood and wild game. The Tarland Food and Music Festival held in September is a great example! Christmas is an exciting time of year in Aberdeen with an array of events in the town and local castles including food markets, craft fairs and live concerts.
Inspired to visit this wonderful region? Hire a motorhome with Goboony to fully explore Aberdeen and what it has to offer!


This is a placeholder page for Vanessa Cano, which means this person is not currently on this site. We do suggest using the tools below to find Vanessa Cano.
You are visiting the placeholder page for Vanessa Cano. This page is here because someone used our placeholder utility to look for Vanessa Cano. We created this page automatically in hopes Vanessa Cano would find it. If you are not Vanessa Cano, but are an alumni of Montclair High School Montclair, CA, register on this site for free now.