text:
Home Sports VAR Drama as Portugal and Spain Reach World Cup Last 16!
VAR Drama as Portugal and Spain Reach World Cup Last 16!
Cristiano Ronaldo's Portugal and 2010 winners Spain progressed to the World Cup knockout rounds on a Monday night of high drama dominated by VAR decisions, as Uruguay pipped Russia to top spot in their group.
Spain ended up topping Group B by virtue of goals scored after an injury-time strike by Iago Aspas — confirmed by the video assistant referee — rescued a 2-2 draw against Morocco in Kaliningrad and will take on the host nation in Moscow on Sunday.
Portugal, who finished level with Spain on five points, conceded a late penalty as Iran drew level at 1-1 and faced a heart-stopping moment as Mehdi Taremi put a shot inch wide — a goal that would have sent the European champions crashing out.
Ronaldo was at the heart of a series of VAR incidents in Saransk. First, he had a penalty saved by Ali Beiranvand and was later saved by a review as Iran's players called for him to be sent off following an incident with Morteza Pouraliganji.
Furious Iran coach Carlos Queiroz said Portugal captain Ronaldo, who was booked, should have been sent off for using an elbow and lashed out against VAR, saying it was "not going well".
"I don't want to talk too much, it's about my country and a player. I know it'll be a war against me," said Queiroz, who coached his native Portugal at the 2010 finals in South Africa."
"But the reality has you stopped the game for VAR, there is an elbow," he added. "An elbow is a red card in the rules. In the rules, it doesn't say if it is (Lionel) Messi or Ronaldo."
Portugal will face Uruguay in the last 16 in Sochi on Saturday.
Relieved Spain boss Fernando Hierro, who took over as the coach on the eve of the World Cup, said he knew the group was going to be tough, acknowledging his side had to tighten up.
"The first conclusion I draw is that despite all these challenges we ended on top. I'm looking at the silver lining," he said.
"Obviously we can improve. Five goals in three matches is not the way forward. The players are professional, they understand what the situation."
– VAR penalties –
There have now been 19 penalties taken at the 2018 World Cup — more than in any other edition of the competition.
Earlier, 10-man Russia were handed a World Cup reality check, losing 3-0 to Uruguay as the South Americans topped Group A with a perfect record.
The host nation rode a wave of national euphoria in their first two matches, sweeping aside Saudi Arabia and Egypt despite being initially written off as no-hopers but could not keep the magic alive against the South Americans.
An early free-kick from Barcelona's Luis Suarez, an own goal from Denis Cheryshev and a late strike from Edinson Cavani sealed top spot for Uruguay, with Russia through to the knockout stages as runners-up.
"Obviously, the first few matches you're tense and don't want to make any mistakes or lose," said man-of-the-match Suarez.
"One mistake can be very expensive. Today we came to the pitch with a different attitude and perspective and that's made the pressure better. That's what we saw on the pitch."
Disappointed Russian coach Stanislav Cherchesov said it was better to lose heavily now than in future games, adding his players had shown nerves for the first time but were psychologically prepared for the knockout stages.
"The outcome is very simple. We did what we had to do," he said. "But you cannot start a game of this level with such a mistake — we gave up a goal.
"Uruguay then started playing their type of football. And then we got a red card."
In the other match in Group A, Saudi Arabia beat Mohamed Salah's Egypt 2-1 in a matchup of two eliminated sides in Volgograd but the main talking point was Egypt goalkeeper Essam el-Hadary becoming the oldest player ever to appear in a World Cup.
El Hadary, 45 years and 161 days old, smashed the record set by Colombia goalkeeper Faryd Mondragon, who was aged 43 years and three days when he played against Japan in Brazil in 2014.
The veteran goalkeeper produced a stunning save from a Fahad al-Muwallad penalty after Salah had given Egypt the lead with his second World Cup goal, but could not keep out another spot kick by Salman al-Faraj in first-half injury time.
World Cup Last 16
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A beautiful Moroccan mirror flag with which even your side mirrors turn in to Moroccan fans.
This product is ideal to combine with our Moroccan carhood flag.
Everyone who sees your car passing by knows that there is a proud lion inside of it!


text:
Q: Selected value in ribbon DropDown/ComboBox is not displayed When I select one of the values in my DropDown (or ComboBox, tried both), the event gets triggered correctly, but the selected item is not shown when the DropDown is closed (the box stays empty). I thought it was a default behaviour to be filled in (like e.g. Language dropdown). I have the problem with both static and dynamically loaded controls.
<DropDown
  Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown"
  Sequence="1"
  Command="MyNS.PageComponent.Command.DoAction"
  Width="150px"
  TemplateAlias="m2">
    <Menu Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown.Menu">
       <MenuSection Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown.MenuSection">
          <Controls Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown.MenuSection.Controls">
             <Button Command="SomeCommand" Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown.Button1" LabelText="Sometext" />
             <Button Command="SomeCommand" Id="MyNS.SharePoint.Ribbon.MyGroup.Dropdown.Button2" LabelText="SomeText" />
          </Controls>
       </MenuSection>
    </Menu>
</DropDown>

A: Unfortunately, all this has to be done manually.
First of all, you will need to provide QueryCommand attribute of DropDown element, and return the correct current value manually to the properties["Value"] variable from the handler script (properties is the second parameter passed to handleCommand function of your ribbon page component).
So you will need some global js variable for storing the current value of the dropdown.
When Button1 or Button2 is pressed, you should update the global variable with the corresponding value. Then, you will have to execute "application state changed" command, to force execution of all the query commands on the page.
"Application state changed" command can be executed with the following code:
SP.Ribbon.PageManager.get_instance().get_commandDispatcher().executeCommand(Commands.CommandIds.ApplicationStateChanged,null);

I've just tested this solution and it works.
P.S. Probably you might be interested to try my opensource project SharePoint 2010 Fluent Ribbon API, which aims to simplify dealing with ribbon and provides MUCH easier interface for this kind of tasks. It supports DropDowns starting from the 1.4 release. I have an example of using DropDown, fetching value from it, etc., in FluentRibbon.Elementary sample project (in RibbonWebPart.cs), which can be found under the [Source Code] tab on CodePlex.

A: You don't need to do anything manually.  You probably have the wrong SelectedItemDisplayMode in your xml.  Try setting it to "Menu".  The default does not display the item.  It will only display something if you set properties.value to something.
          <DropDown
            Id="..."
            Command="..."
            SelectedItemDisplayMode="Menu"

redpajama_set_name: RedPajamaStackExchange


text:
Q: Using "chmod -R 666 filename" does not give all files writing permissions I have been running as root in a Linux Virtual Machine to change permissions to a whole list of files within a directory. When I check the permissions after using sudo chmod 666 -R /home/candidate/working/other_files, all the files have read and write privileges besides one hidden file (highlighted in the image). I am not sure how I can make it so that file will also have write privileges when I specifically wrote the chmod command that should give all the files read and write privileges.
The file .. does not have write permission for all other users after using chmod:
root@d3ac9495a31:/home/candidate/working# chmod -R 666 /home/candidate/working/other_files/
root@d3ac9495a31:/home/candidate/working# ls -la other_files
total 14108
drw-rw-rw- 1 root      root         4096 Feb 27  2018 .
drwxrwxr-x 1 candidate candidate    4096 Feb 27  2018 ..
-rw-rw-rw- 1 root      root      4537584 Feb 27  2018 file1.html
drwxrwxr-x 1 root      root      1123786 Feb 27  2018 file2.html
-rw-rw-rw- 1 root      root      5747804 Feb 27  2018 file3.html
-rw-rw-rw- 1 root      root      3013094 Feb 27  2018 file4.html

A: There are a number of issues here

*

*Technical. You have removed search permission (x) for the other_files directory (represented in the listing as .). This will cause problems when you try to list it, or access files in it. Fix with chmod a+x other_files

*Convenience. You don't always need to specify a full filesystem path for chmod. Here, since you are already in working you could have used chmod -R 666 other_files. But see #1 as this is actually wrong - you should only be changing permissions of the files and not the directory too. So chmod 666 other_files/* would have been better and more correct.

*Understanding. The . directory entry (see the leading d in the permissions block within ls -l) represents the current directory, in this case other_files. The .. entry points to the parent directory, in this case working. Parent directories are typically excluded from recursive operations as otherwise you'd end up travelling up the filesystem tree as well as down.

*Operational. Generally, you shouldn't really be working as root when you're in candidate's home directory. Consider resetting the files' owner so that it's candidate, and then work as candidate instead of root. For starters, this will make it harder for you to break your system accidentally.


I smiled bitterly as I wiped the tears rolling down my cheeks, sitting in the dark corner of my room.
No fake smiles, no unsincere 'I'm okay', no pretentious act, nothing but the blank atmosphere. My empty world.
Holding my phone tightly with my shivering hand, I speak the thoughts that I locked  in my heart for 2 years and finally releasing it.
"sorry" I whispered "sorry for breaking my promise"
"I promised to be  strong... I promised to be brave..I promised " I bit my lower lip to suppress my sob "I promised to fight for you--for us"
But the feeling is too much..that it could still break my already shattered heart.
"I-I love you" I sob.
"I love you so much that it hurts.. It hurts so much baby, God knows how much I wanted to fight still..."
"but it has to end. We have fought enough Dan... I've fought e-enough"
I need to let us go.


text:
a synthetic consensus anti-spike protein dna vaccine induces protective immunity against middle east respiratory syndrome coronavirus in nonhuman primates. first identified in 2012, middle east respiratory syndrome (mers) is caused by an emerging human coronavirus, which is distinct from the severe acute respiratory syndrome coronavirus (sars-cov), and represents a novel member of the lineage c betacoronoviruses. since its identification, mers coronavirus (mers-cov) has been linked to more than 1372 infections manifesting with severe morbidity and, often, mortality (about 495 deaths) in the arabian peninsula, europe, and, most recently, the united ... 0 26290414
tropism and replication of middle east respiratory syndrome coronavirus from dromedary camels in the human respiratory tract: an in-vitro and ex-vivo study. middle east respiratory syndrome coronavirus (mers-cov) is a zoonotic infection causing severe viral pneumonia, with index cases having resided in or recently travelled to the arabian peninsula, and is a global concern for public health. limited human-to-human transmission, leading to some case clusters, has been reported. mers-cov has been reported in dromedary camels but phenotypic characterisation of such viruses is limited. we aimed to compare mers-cov isolates from dromedaries in saudi arab ... 2014 25174549
host species restriction of middle east respiratory syndrome coronavirus through its receptor, dipeptidyl peptidase 4. middle east respiratory syndrome coronavirus (mers-cov) emerged in 2012. recently, the mers-cov receptor dipeptidyl peptidase 4 (dpp4) was identified and the specific interaction of the receptor-binding domain (rbd) of mers-cov spike protein and dpp4 was determined by crystallography. animal studies identified rhesus macaques but not hamsters, ferrets, or mice to be susceptible for mers-cov. here, we investigated the role of dpp4 in this observed species tropism. cell lines of human and nonhuman ... 2014 24899185
comparative pathology of rhesus macaque and common marmoset animal models with middle east respiratory syndrome coronavirus. middle east respiratory syndrome (mers), which is caused by a newly discovered coronavirus (cov), has recently emerged. it causes severe viral pneumonia and is associated with a high fatality rate. however, the pathogenesis, comparative pathology and inflammatory cell response of rhesus macaques and common marmosets experimentally infected with mers-cov are unknown. we describe the histopathological, immunohistochemical, and ultrastructural findings from rhesus macaque and common marmoset animal ... 2017 28234937
an acute immune response to middle east respiratory syndrome coronavirus replication contributes to viral pathogenicity. middle east respiratory syndrome coronavirus (mers-cov) was first identified in a human with severe pneumonia in 2012. since then, infections have been detected in >1500 individuals, with disease severity ranging from asymptomatic to severe, fatal pneumonia. to elucidate the pathogenesis of this virus and investigate mechanisms underlying disease severity variation in the absence of autopsy data, a rhesus macaque and common marmoset model of mers-cov disease were analyzed. rhesus macaques develo ... 2016 26724387
animal models of middle east respiratory syndrome coronavirus infection. the emergence of the middle east respiratory syndrome coronavirus (mers-cov) in 2012 marked the second time that a new, highly pathogenic coronavirus has emerged in the human population in the 21st century. in this review, we discuss the current state of knowledge of animal models of mers-cov infection. commonly used laboratory animal species such as syrian hamsters, mice and ferrets are not susceptible to mers-cov, due to differences in the mers-cov receptor dipeptidyl peptidase 4 (dpp4). the i ... 2015 26192750
infection with mers-cov causes lethal pneumonia in the common marmoset. the availability of a robust disease model is essential for the development of countermeasures for middle east respiratory syndrome coronavirus (mers-cov). while a rhesus macaque model of mers-cov has been established, the lack of uniform, severe disease in this model complicates the analysis of countermeasure studies. modeling of the interaction between the mers-cov spike glycoprotein and its receptor dipeptidyl peptidase 4 predicted comparable interaction energies in common marmosets and human ... 2014 25144235
a comparative review of animal models of middle east respiratory syndrome coronavirus infection. middle east respiratory syndrome coronavirus (mers-cov) was initially isolated from a saudi arabian man with fatal pneumonia. since the original case in 2012, mers-cov infections have been reported in >1500 humans, and the case fatality rate is currently 35%. this lineage c betacoronavirus has been reported to cause a wide range of disease severity in humans, ranging from asymptomatic to progressive fatal pneumonia that may be accompanied by renal or multiorgan failure. although the clinical pre ... 2016 26869154
middle east respiratory syndrome coronavirus (mers-cov) causes transient lower respiratory tract infection in rhesus macaques. in 2012, a novel betacoronavirus, designated middle east respiratory syndrome coronavirus or mers-cov and associated with severe respiratory disease in humans, emerged in the arabian peninsula. to date, 108 human cases have been reported, including cases of human-to-human transmission. the availability of an animal disease model is essential for understanding pathogenesis and developing effective countermeasures. upon a combination of intratracheal, ocular, oral, and intranasal inoculation with ... 2013 24062443
[advances in the animal models of mers-cov]. the middle east respiratory syndrome coronavirus (mers-cov) was first isolated in 2012 from patients that died from severe pneumonia. it was another coronavirus which resulted in severe infection of humans, apart from the severe acute respiratory syndrome coronavirus. the development of an appropriate animal model is necessary to study the pathogenesis and to evaluate the intervening measures against mers-cov infection. to date, several small animals(e.g., mice, syrian hamsters, and ferrets) hav ... 2016 29963837
a spike-modified middle east respiratory syndrome coronavirus (mers-cov) infectious clone elicits mild respiratory disease in infected rhesus macaques. the recurrence of new human cases of middle east respiratory syndrome coronavirus (mers-cov) underscores the need for effective therapeutic countermeasures. nonhuman primate models are considered the gold standard for preclinical evaluation of therapeutic countermeasures. however, mers-cov-induced severe respiratory disease in humans is associated with high viral loads in the lower respiratory tract, which may be difficult to achieve in nonhuman primate models. considering this limitation, we wa ... 2018 30013082
a recombinant vsv-vectored mers-cov vaccine induces neutralizing antibody and t cell responses in rhesus monkeys after single dose immunization. middle east respiratory syndrome coronavirus (mers-cov) has been a highly threatening zoonotic pathogen since its outbreak in 2012. similar to sars-cov, mers-cov belongs to the coronavirus family and can induce severe respiratory symptoms in humans, with an average case fatality rate of 35% according to the world health organization. spike (s) protein of mers-cov is immunogenic and can induce neutralizing antibodies, thus is a potential major target for vaccine development. here we constructed a ... 2018 29246504
treatment with interferon-α2b and ribavirin improves outcome in mers-cov-infected rhesus macaques. the emergence of middle east respiratory syndrome coronavirus (mers-cov) is of global concern: the virus has caused severe respiratory illness, with 111 confirmed cases and 52 deaths at the time of this article's publication. therapeutic interventions have not been evaluated in vivo; thus, patient management relies exclusively on supportive care, which, given the high case-fatality rate, is not highly effective. the rhesus macaque is the only known model organism for mers-cov infection, developi ... 2013 24013700
interferon-β and mycophenolic acid are potent inhibitors of middle east respiratory syndrome coronavirus in cell-based assays. the middle east respiratory syndrome coronavirus (mers-cov) presents a novel emerging threat to public health worldwide. several treatments for infected individuals have been suggested including ifn, ribavirin and passive immunotherapy with convalescent plasma. administration of ifn-α2b and ribavirin has improved outcomes of mers-cov infection in rhesus macaques when administered within 8 h post-challenge. however, detailed and systematic evidence on the activity of other clinically available dr ... 2014 24323636
evaluation of candidate vaccine approaches for mers-cov. the emergence of middle east respiratory syndrome coronavirus (mers-cov) as a cause of severe respiratory disease highlights the need for effective approaches to cov vaccine development. efforts focused solely on the receptor-binding domain (rbd) of the viral spike (s) glycoprotein may not optimize neutralizing antibody (nab) responses. here we show that immunogens based on full-length s dna and s1 subunit protein elicit robust serum-neutralizing activity against several mers-cov strains in mice ... 2015 26218507
recombinant receptor binding domain protein induces partial protective immunity in rhesus macaques against middle east respiratory syndrome coronavirus challenge. development an effective vaccine against middle east respiratory syndrome coronavirus (mers-cov) is urgent and limited information is available on vaccination in nonhuman primate (nhp) model. we herein report of evaluating a recombinant receptor-binding domain (rrbd) protein vaccine in a rhesus macaque model. 2015 26629538
3b11-n, a monoclonal antibody against mers-cov, reduces lung pathology in rhesus monkeys following intratracheal inoculation of mers-cov jordan-n3/2012. middle east respiratory syndrome coronavirus (mers-cov) was identified in 2012 as the causative agent of a severe, lethal respiratory disease occurring across several countries in the middle east. to date there have been over 1600 laboratory confirmed cases of mers-cov in 26 countries with a case fatality rate of 36%. given the endemic region, it is possible that mers-cov could spread during the annual hajj pilgrimage, necessitating countermeasure development. in this report, we describe the cli ... 2016 26828465
mers-cov virus-like particles produced in insect cells induce specific humoural and cellular imminity in rhesus macaques. middle east respiratory syndrome coronavirus (mers-cov) causes severe respiratory disease in humans with a case fatality rate of over 39%, and poses a considerable threat to public health. a lack of approved vaccine or drugs currently constitutes a roadblock in controlling disease outbreak and spread. in this study, we generated mers-cov vlps using the baculovirus expression system. electron microscopy and immunoelectron microscopy results demonstrate that mers-cov vlps are structurally similar ... 2017 27050368
generation of a transgenic mouse model of middle east respiratory syndrome coronavirus infection and disease. the emergence of middle east respiratory syndrome-coronavirus (mers-cov) in the middle east since 2012 has caused more than 900 human infections with ∼40% mortality to date. animal models are needed for studying pathogenesis and for development of preventive and therapeutic agents against mers-cov infection. nonhuman primates (rhesus macaques and marmosets) are expensive models of limited availability. although a mouse lung infection model has been described using adenovirus vectors expressing h ... 2015 25589660
a single dose of chadox1 mers provides protective immunity in rhesus macaques. developing a vaccine to protect against the lethal effects of the many strains of coronavirus is critical given the current global pandemic. for middle east respiratory syndrome coronavirus (mers-cov), we show that rhesus macaques seroconverted rapidly after a single intramuscular vaccination with chadox1 mers. the vaccine protected against respiratory injury and pneumonia and reduced viral load in lung tissue by several orders of magnitude. mers-cov replication in type i and ii pneumocytes of c ... 2020 32577525
clinical benefit of remdesivir in rhesus macaques infected with sars-cov-2. effective therapies to treat coronavirus disease 2019 (covid-19) are urgently needed. while many investigational, approved, and repurposed drugs have been suggested as potential treatments, preclinical data from animal models can guide the search for effective treatments by ruling out those that lack efficacy in vivo. remdesivir (gs-5734) is a nucleotide analogue prodrug with broad antiviral activity1,2 that is currently being investigated in covid-19 clinical trials and recently received emerge ... 2020 32516797
importance of neutralizing monoclonal antibodies targeting multiple antigenic sites on mers-cov spike to avoid neutralization escape. middle east respiratory syndrome coronavirus (mers-cov) causes a highly lethal pulmonary infection with ∼35% mortality. the potential for a future pandemic originating from animal reservoirs or healthcare-associated events is a major public health concern. there are no vaccines or therapeutic agents currently available for mers-cov. using a probe-based single b cell-cloning strategy, we have identified and characterized multiple neutralizing mabs specifically binding to the receptor binding doma ... 2018 29514901
pathogenicity and viral shedding of mers-cov in immunocompromised rhesus macaques. middle east respiratory syndrome coronavirus (mers-cov) has recently emerged in the middle east. since 2012, there have been approximately 2,100 confirmed cases, with a 35% case fatality rate. disease severity has been linked to patient health status, as people with chronic diseases or an immunocompromised status fare worse, although the mechanisms of disease have yet to be elucidated. we used the rhesus macaque model of mild mers to investigate whether the immune response plays a role in the pa ... 2018 29483914
Displaying items 1 - 23 of 23

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Laravel: How to make a relationship between two table that has a reference table I have 3 tables.
People, Assignments and Projects

each person in People table can be assigned to one or more Projects.
  Each project in Projects table can have one ore more person assigned to it.

Here are the columns of each table:
People

*

*id

*name

Assignments:

*

*person_id

*project_id

Projects

*

*id

*title

My question is: in my Project model, how do I define the relationship?
What I want is to be able to get the names of all person assigned to a specific project.
I NEED:
$project->assignedPeople

So I can:
foreach($projects as $project)
$project->title
  foreach(projects->assignedPeople as $person)
  $person->name

I NEED YOUR HELP PLEASE
THANK YOU!

A: Actually You need Many-to-Many relationship.
You can define the Models like this
Project Model
class Project extends Model {

    public function peoples()
    {
        return $this->belongsToMany('App\People');
    }
}

And
People Model
class People extends Model {

    public function projects()
    {
        return $this->belongsToMany('App\Project');
    }
}

Now you can access the names of all person assigned to a specific project like this.
$project->peoples

And you can run your foreach loop like this
foreach($projects as $project) {
    echo $project->title;
    foreach(project->peoples as $person) {
        echo $person->name;
    }
}

Hope it may solve your problem.
Thanks

A: // Inside Model Project
class Project extends Model {

    public function people()
    {
        return $this->hasMany('App\Models\People');
    }

}

// Inside Model People
class People extends Model {

    public function project()
    {
        return $this->belongsToMany('App\Models\Project');
    }

}

// In case of custom joining table:
class People extends Model {

    public function project()
    {
         return $this->belongsToMany('App\Models\Project', 'Assignments', 'person_id', 'project_id');
    }

}


Kriti Sanon has had a busy weekend in Delhi. Doing the promotional rounds for her upcoming feature Bareilly Ki Barfi opposite stars Rajkummar Rao and Ayushmann Khurrana, the star was spotted on multiple occasions in the city.
Sanon's hectic schedule demanded several wardrobe changes, but if we could pick one favourite look from the weekend, it would be the sunny dress by label Ka Sha. The tie-dye pattern of the lemon-yellow dress-and-jacket combo was offset with contrast striped detailing, which made for a visually interesting ensemble. The star accessorised the look with white sneakers, black sunglasses and silver jewellery by Ritika Sachdeva.
In the uncertain presence of sunlight in the monsoon, Sanon's choice of colour injects the perfect dose of optimism on a gloomy day. Scroll through our gallery to get a closer look at the star's ensemble.


text:
Home > Theses, Dissertations and Capstone Projects > Education > 638
Keeping the peace: Conflict resolution techniques in a sixth grade classroom
Meryl Cherry, Pacific University
Thesis (On-Campus Access Only)
Carol Morgaine
The classroom is a place where children develop an understanding about relationships among people and how people act when conflict arises. The purpose of this paper was to investigate the conflict resolution techniques children were being exposed to in a sixth grade classroom at Elmonica Elementary School. Deutsch's (1973) theory of constructive and destructive conflict resolution guided the examination of conflict resolution techniques used in the classroom.
The classroom was studied for four days over the course of a month. The paper included narrative accounts of what was observed in the classroom during that time period. The investigation revealed conflicts in the classroom were resolved destructively as a result of competition, intolerance, poor communication, inappropriate expression of emotion, a lack of conflict resolution skills and, misuse of power by the teacher. It was also found the teacher failed to model constructive conflict resolution techniques. In addition, there was not an integration of conflict resolution skills into the curriculum.
Implications which emerged from the study were that teaching conflict management needed to be implemented holistically with the teacher serving as a role model for resolving conflict constructively. It was also found conflict resolution was a learned skill. The skill required the students to be able to work together in cooperation, communicate positively, identify and discuss their feelings, demonstrate a tolerance: for diversity, identify conflict, and devise plans for resolving conflict. Students needed opportunities to rehearse the skills and to be provided with feedback about their effectiveness in using the skills.
Cherry, Meryl, "Keeping the peace: Conflict resolution techniques in a sixth grade classroom" (1993). College of Education. 638.
https://commons.pacificu.edu/educ/638
Collection Policies


text:
Пролетар — найманий робітник, позбавлений засобів виробництва.

 Пролетар

 У Стародавньому Римі — громадянин, який не мав майнового цензу.
 Найманий робітник.
 заст. Людина, що не має власності.
 Пролетар — робітнича газета 1930-х років
 Пролетар — колишня назва видавництва «Політвидав України»

Села:
 Пролетар — колишня назва села Дубина (Вінницька область, Хмільницький район)
 Пролетар — колишня назва села Степне (Полтавська область, Лохвицький район)
Селище:
 Пролетар — колишня назва селища Малі Феськи (Харківська область, Золочівський район)
Місцевості
 Пролетар — місцевість і цвинтар у Донецьку

Див. також
 Пролетарське
 Пролетарка
 Пролетарій
 Пролетарська вулиця
 Пролетарський район

redpajama_set_name: RedPajamaWikipedia


text:
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text:
\section{Introduction}
The classical Brill-Noether theorem states that if we are given
$g,r,d \geq 0$, a general curve $X$ of genus $g$ carries a linear series
$({\mathscr L},V)$ of projective dimension $r$ and degree $d$ if and only if the quantity
$$\rho(g,r,d):=g-(r+1)(r+g-d)$$
is nonnegative \cite{g-h1}. Moreover, in this case
the moduli space $G^r_d(X)$ of such linear series has pure dimension
$\rho$. This statement was generalized by Eisenbud and Harris
to allow for imposed ramification: given marked points
$P_1,\dots,P_n \in X$, and sequences $0\le a^i_0<\dots<a^i_r \leq d$ for
$i=1,\dots,n$, consider the moduli space
$G^r_d(X,(P_1,a^1_{\bullet}),\dots,(P_n,a^n_{\bullet})) \subseteq
G^r_d(X)$ parametrizing linear series with vanishing sequence at least
$a^i_{\bullet}$ at each of the $P_i$. Then Eisenbud and Harris used their
theory of limit linear series to show in \cite{e-h1}
that in characteristic $0$, if $(X,P_1,\dots,P_n)$ is a general $n$-marked
curve of genus $g$, the dimension of
$G^r_d(X,(P_1,a^1_{\bullet}),\dots,(P_n,a^n_{\bullet}))$---if it is
nonempty---is given by the generalized formula
$$\rho(g,r,d,a^1_\bullet,\ldots,a^n_\bullet):=g-(r+1)(r+g-d)-\sum_{i=1}^n \sum_{j=0}^r (a^i_j-j).$$
The condition for nonemptiness is still combinatorial, but becomes more
complicated in this context.

This theorem fails in positive characteristic
for $n \geq 3$, but is still true if $n \leq 2$.  In this case, we also
have a simple criterion for nonemptiness. To state it, we shift
notation, supposing we have marked points $P,Q\in X$, and sequences
$a_{\bullet}$, $b_{\bullet}$. We then introduce the following notation:
$$\widehat{\rho}(g,r,d,a_\bullet,b_\bullet) :=
g-\sum_{j:a_j+b_{r-j} > d-g} a_j+b_{r-j}-(d-g).$$

We summarize what was previously known about the space
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$, as follows.

\begin{thm}\label{thm:bg}
Given $(g,r,d)$ nonnegative integers,
and sequences $0 \leq a_0<a_1 <\dots<a_r \leq d$,
$0 \leq b_0<b_1 <\dots<b_r \leq d$, let $(X,P,Q)$ be a twice-marked
smooth projective curve of genus $g$ over a field of any characteristic.
Set $\rho=\rho(g,r,d,a_{\bullet},b_{\bullet})$ and
set $\widehat{\rho}=\widehat{\rho}(g,r,d,a_{\bullet},b_{\bullet})$.

Suppose that $X$ and $P,Q$ are general. Then
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is nonempty if and only if
$\widehat{\rho} \geq 0$, and if nonempty, has pure dimension
$\rho$. Furthermore, it is reduced and Cohen-Macaulay, and if
$\widehat{\rho} \geq 1$, it is connected.
\end{thm}

For the nonemptiness and dimension statements, see \cite{os18};
for reducedness and connectedness, see \cite{os26}. The Cohen-Macaulayness
statement follows from the construction of
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ (see for instance the proof
of Proposition \ref{prop:non-smooth} below) together with the
Cohen-Macaulayness of relative Schubert cycles.

What has remained open until now is the question of the singularities
of the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$. In the absence of marked points,
Gieseker in 1982 used degenerations to prove a conjecture of
Petri that if $X$ is general, then the space $G^r_d(X)$ is also smooth
\cite{gi1}. This proof was later simplified by Eisenbud and Harris
\cite{e-h2} and Welters \cite{we4} using ideas closely related to the
theory of limit linear series. These proofs all relied on proving injectivity
of the
Petri map, by taking a hypothetical nonzero element of the kernel, and
carrying out a careful analysis of how it would behave under degeneration.

In this paper, we give a new proof of the Gieseker-Petri theorem, and
generalize it to the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$,
proving that the singular locus of this space consists precisely of linear series with a certain type of excess vanishing.
Our Gieseker-Petri theorem with imposed ramification, Theorem~\ref{thm:main} below, generalizes two statements.
\begin{enumerate}
\item In the absence of marked points, it reduces to the Gieseker Petri theorem, which holds for curves of any genus.
\item With marked points allowed, but in genus $0$, it reduces to the well-known characterization of the singular loci of Schubert varieties and Richardson varieties.
\end{enumerate}

Indeed, in the case $g=0$, a single ramification condition corresponds to a
Schubert cycle in the Grassmannian $\mathrm{Gr}(r+1,\mathcal{O}_{\mathbb{P}^1}\!(d))$, while a pair of ramification
conditions similarly corresponds to a Richardson variety. These spaces are
singular, and their singularities can be characterized precisely as
loci with a specific type of excess vanishing. Our main theorem
extends this characterization to all genera, and also deduces additional
consequences on the geometry of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$.
To state it, the following preliminary notation will be helpful.

\begin{notn} If $({\mathscr L},V)$ is a ${\mathfrak g}^r_d$ on a smooth projective curve
$X$, and $D$ is an effective divisor on $X$, write
$$V(-D)=V \cap \Gamma(X,{\mathscr L}(-D)) \subseteq \Gamma(X,{\mathscr L}).$$
\end{notn}

Thus, to say that $({\mathscr L},V)$ has vanishing sequence at least $a_{\bullet}$
at $P$ is equivalent to saying that
\begin{equation}\label{eq:van-ineq}
\dim V(-a_j P) \geq r+1-j
\end{equation}
for $j=0,\dots,r$.

We then make the following definition:

\begin{defn}\label{defn:good-open} In the situation of Theorem
\ref{thm:bg}, let
$$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \subseteq G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$$
be the open subset consisting of $({\mathscr L},V)$ such that
\eqref{eq:van-ineq} holds with equality for all $j>0$ such that
$a_j>a_{j-1}+1$, and the analogous condition holds for  $(Q,b_\bullet)$.
\end{defn}

We see that $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$
contains all linear
series with precisely the prescribed vanishing at $P$ and $Q$, but it
also contains many linear series with more than the prescribed vanishing.
For instance, if $a_{\bullet} = b_{\bullet} = (0,1,\ldots,r)$ are both minimal, so that
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))=G^r_d(X)$, then we also have
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))=G^r_d(X)$.
Our main theorem is then the following.

\begin{thm}\label{thm:main} In the situation of Theorem \ref{thm:bg},
suppose further that we are in characteristic $0$. Then the smooth locus of
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is precisely equal to
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$.

Furthermore, the
space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ has singularities in
codimension at least $3$, is normal, and when $\widehat{\rho} \geq 1$
is irreducible.
\end{thm}

Thus, we are in particular giving a new proof of the Gieseker-Petri
theorem (in characteristic $0$).
As an immediate consequence of Theorem~\ref{thm:main}, the twice-pointed Brill-Noether curves studied in \cite{c-l-p-t1}, as well as twice-pointed Brill-Noether surfaces \cite{c-p3, a-c-t1} are smooth.

Our proof proceeds by degenerating to a chain of elliptic curves,
and studying the geometry of the corresponding moduli space of
Eisenbud-Harris limit linear series.
The key
idea in this step is that although the space of limit linear series
will be singular in codimension $1$, after base change and blowup one can
ensure that any given point of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ on the generic fiber will
specialize to a smooth point of the limit linear series space of a chain of curves of genus 0 or 1. This
is where the characteristic-$0$ hypothesis comes in. The case of genus 0 is well-known, so our main
calculation is  the following result, which does not depend on
characteristic, concerning the case $g=1$.

\begin{thm}\label{thm:genus-1} In the situation of Theorem \ref{thm:bg},
suppose that $g=1$, and make the generality condition explicit as follows:
$X$ is arbitrary, and $P,Q$ are such that $P-Q$ is not a torsion
point of $\operatorname{Pic}^0(X)$ of order less than or equal to $d$.
Then the space $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth.
\end{thm}

The proof of Theorem \ref{thm:genus-1} proceeds by consideration of
the morphism
\begin{equation}\label{eq:to-pic}
G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \operatorname{Pic}^d(X).
\end{equation}
The main subtlety that needs to be addressed is that the map~\eqref{eq:to-pic}
is not smooth. The fibers are each described as an intersection of a pair
of Schubert cycles in a Grassmannian.  But in finitely many fibers, namely the ones above line bundles of the form $\mathcal{O}_X(aP+(d\!-\!a)Q)$ for $0<a<d$, the
pairs of flags defining the Schubert cycles are not transverse, but only almost-transverse (see
Definition~\ref{def:a-t}). We prove Theorem
\ref{thm:genus-1} by first showing that in fibers, the tangent spaces
have dimension at most $1$ greater than expected, and then showing
that at the points where the tangent space dimension jumps in the fiber,
there cannot be any horizontal tangent vectors.

The statement on tangent spaces in fibers,
which is Corollary \ref{cor:at} below, takes place entirely inside the
Grassmannian, and may be of independent interest. Indeed, \S
\ref{sec:a-t} is a study of tangent spaces of intersections of
pairs of Schubert cycles, and we address the case of arbitrary pairs of
flags in Theorem \ref{thm:workhorse} and Remark \ref{rem:arbitrary-flags}.
We then prove Theorems \ref{thm:main} and \ref{thm:genus-1} in \S
\ref{sec:proofs}.

\section{Almost-transverse intersections of Schubert cycles}\label{sec:a-t}

It is well known that the intersection of two Schubert varieties associated
to transverse flags---commonly called a Richardson variety---is
smooth on the open subset of points which are smooth in both Schubert
varieties.  In this section we consider intersections of pairs of Schubert
varieties associated to not necessarily transverse flags. Our analysis
recovers the usual smoothness statement in the transverse case, but our main
purpose is to analyze the almost-transverse case in Corollary \ref{cor:at},
where we characterize the smooth
points and show that the dimension of the tangent space jumps only by $1$
at the non-smooth points. While Schubert intersections and non-transverse
flags have been studied by Vakil \cite{va7} and Coskun
\cite{co3}, those situations involved studying the flat limits of transverse
intersections, rather than the direct analysis of the non-transverse
intersections required in the present work.\footnote{More precisely, they
work with closures of loci with prescribed behavior with respect to both
flags; these are in particular irreducible, so are not the same thing as
the intersection of Schubert cycles which we consider.}

We fix $k$ to be an algebraically closed field of any characteristic.
Throughout this section, we will work entirely with $k$-valued (equivalently,
closed) points.
We index our complete flags by codimension, so that for a complete flag $P^\bullet$ in a $d$-dimensional vector space $H$,
\begin{align*}
0 &= P^d \subset \cdots \subset P^1 \subset P^0 = H.
\end{align*} We fix further notation as
follows.

\begin{defn}\label{defn:schubert}
Given a $k$-vector space $H$ of finite dimension $d$ and
a complete flag $P^{\bullet}$ in $H$, if we are given also
$a_{\bullet} = (a_0,\ldots,a_r)\in {\mathbb Z}^{r+1}$
with $$0 \le a_0 < \cdots < a_r < d,$$
we let $\Sigma_{P^{\bullet}, a_{\bullet}}$ be the \textbf{Schubert variety}
defined as the closed subscheme of $\operatorname{Gr}(r+1,H)$ given by the set of
$\Lambda \in \operatorname{Gr}(r+1,H)$ such that
\begin{equation}\label{eq:one-flag}
\dim (\Lambda\cap P^{a_i})
\ge r+1-i
\end{equation}
for $i=0,\ldots,r$.
\end{defn}

More precisely, the conditions in~\eqref{eq:one-flag} are determinantal,
yielding a scheme structure on $\Sigma_{P^{\bullet},a_{\bullet}}$ (which turns out
to be reduced). In our notation, the codimension of $\Sigma_{P^{\bullet},a_{\bullet}}$
is given by $\sum_{i=0}^r (a_i\!-\!i)$.

\begin{defn}\label{defn:active}
With $a_{\bullet} = (a_0,\ldots , a_r)$ an increasing sequence as above, say that an index $i$ with $0\le i\le r$ is \textbf{active} in $a_{\bullet}$ if $i>0$ and
$a_i > a_{i-1} + 1$, or $i=0$ and $a_0 >0.$
\end{defn}

\begin{defn}\label{defn:sigma-circ}
Let ${\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$ be the open
subscheme of
$\Sigma_{P^{\bullet},a_{\bullet}}$ consisting of subspaces $\Lambda$ for which for every active index $i$, the
inequality in~\eqref{eq:one-flag} is an equality.
\end{defn}

\noindent Note that \eqref{eq:one-flag} is automatically an equality when $i=0$, so
in Definition \ref{defn:sigma-circ} we can restrict to positive active
choices of $i$.

We fix the following situation throughout this section.

\begin{sit}\label{sit:flags}
Let $H$ be a finite-dimensional $k$-vector space, and write $d:=\dim H$. Fix complete flags
$P^{\bullet}$, $Q^{\bullet}$ in $H$, and sequences
$a_{\bullet},b_{\bullet}\in {\mathbb Z}^{r+1}$
with $0 \le a_0 < \cdots < a_r < d$ and $0 \le b_0 < \cdots < b_r < d.$
\end{sit}

Recall that we are indexing by codimension; thus $\operatorname{codim} P^i=\operatorname{codim} Q^i=i$.
Note that for any $\Lambda \in {\Sigma}_{P^{\bullet},a_{\bullet}}$, the distinct subspaces in the collection
$\Lambda\cap P^{j}$ form a complete flag in $\Lambda$;
we denote the flag $\Lambda\cap P^{\bullet}$ by abuse of notation.

We have the following description of the tangent space at any point in
${\Sigma}_{P^{\bullet},a_{\bullet}}$. Tangent spaces to Schubert varieties are well
understood \cite{b-l2}, but for the sake of completeness, we provide a
description in the particular case that we need of Grassmannian Schubert
varieties.

\begin{prop}
\label{prop:tangent-schubert}
\mbox{}
\begin{enumerate}
\item Given
$\Lambda \in {\Sigma}_{P^{\bullet},a_{\bullet}}$, let $S$ be the set of active indices $i$ such that $\dim \Lambda\cap P^{a_i} = r+1-i$.  Then
there is a canonical isomorphism of vector spaces
$$T_{\Lambda}\Sigma_{P^{\bullet},a_{\bullet}}\cong
\left\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i})
\subseteq (P^{a_i}+\Lambda)/\Lambda \text{ for } i\in S\right\}.$$

\item In particular, the smooth locus of $\Sigma_{P^{\bullet},a_{\bullet}}$ is precisely
${\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$.
\end{enumerate}
\end{prop}

\begin{proof}
By definition, $\Sigma_{P^{\bullet},a_{\bullet}}$ is the scheme-theoretic intersection
of the following subschemes of $\operatorname{Gr}(r+1,H)$ (for $i=0,1,\cdots,r$):
$$\Sigma_i = \{ \Lambda \in \operatorname{Gr}(r+1,H)\colon \dim(\Lambda \cap P^{a_i}) \geq
r+1-i \}.$$
Define $\Sigma_i^\circ$ to be the open subscheme of $\Sigma_i$ where equality
holds. Note also that in fact
$\Sigma_{P^{\bullet},a_{\bullet}}$ can be cut out as the intersection of the
$\Sigma_i$ over all active indices $i$: this is immediate set-theoretically,
and is also true scheme-theoretically because whenever $a_{i+1}=a_i+1$,
the condition for $a_{i+1}$ is obtained from that of $a_i$ by adding a
single row to the local matrix expression, and considering minors of
size one larger. Thus, every minor occuring in the $a_{i+1}$ condition can be
expanded in terms of minors occuring in the $a_i$ condition.

The first statement of the proposition then follows immediately from the
following claim. For a fixed index $i$,
$$ T_{\Lambda} \Sigma_i =
\begin{cases}
\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i}) \subseteq
(P^{a_i}+\Lambda)/\Lambda \} & \mbox{ if $\Lambda \in \Sigma^\circ_i$}\\
T_{\Lambda} \operatorname{Gr}(r+1,H) & \mbox{ otherwise,}
\end{cases}$$
where we identify $T_\Lambda \operatorname{Gr}(r+1,H)$ with $\operatorname{Hom}(\Lambda,H/\Lambda)$ as
usual.

To prove this claim, one may work on an affine open subset of $\operatorname{Gr}(r+1,H)$, as
follows. Choose a basis of $H$ extending a basis of $\Lambda$; then an affine
neighborhood of $\Lambda$ is given by the set of $(r+1) \times d$ matrices
whose first $(r+1)$ columns form the identity matrix (where the point in
$\operatorname{Gr}(r+1,H)$ is given by taking the span of the rows). More precisely, for any
$k$-algebra $R$, we may identify the $R$-points of this open subscheme with
$R$-valued matrices whose first $r+1$ columns form the identity matrix. In
particular, taking $R = k[\epsilon]$, the tangent space $T_{\Lambda}
\operatorname{Gr}(r+1,H)$ is identified with matrices in block form $( I\  \epsilon M)$,
where $M$ is a matrix of values of $k$; the matrix $M$ then determines an
element of $\operatorname{Hom}(\Lambda,H / \Lambda)$. Now, we may further assume that the
chosen basis of $H$ also includes a basis of $P^{a_i}$ as a subset. Then the
$R$-points of $\Sigma_i$ consist of those matrices such that the submatrix
consisting of all columns not corresponding to the basis of $P^{a_i}$ has rank
at most $i$. Assuming that we order our basis of $\Lambda$ so that a basis of $\Lambda
\cap P^{a_i}$ comes at the end, the submatrix in
question has the form
$$\begin{pmatrix} I & A \\ 0 & B \end{pmatrix},$$
where the size of the identity matrix in the upper left is $\dim (\Lambda /
(\Lambda \cap P^{a_i}))$. Therefore, lying in $\Sigma_i$ corresponds to the
condition that
$$\operatorname{rk}(B) \leq i - (r+1) + \dim (\Lambda \cap P^{a_i}).$$
Now, specialize to the case $R = k[\epsilon]$, and consider a tangent vector to
$\operatorname{Gr}(r+1,H)$ at $\Lambda$. The submatrix $B$ is a multiple of $\epsilon$.
Therefore all $2 \times 2$ and larger minors of $B$ are guaranteed to vanish.
Thus in the case $\dim (\Lambda \cap P^{a_i}) > r+1-i$ (i.e. $\Lambda \not\in
\Sigma_i^\circ$), all tangent vectors to $\operatorname{Gr}(r+1,H)$ at $\Lambda$ are also
tangent vectors to $\Sigma_i$ at $\Lambda$. On the other hand, when $\Lambda
\in \Sigma_i^\circ$, a tangent vector to $\operatorname{Gr}(r+1,H)$ at $\Lambda$ is a tangent
vector to $\Sigma_i$ if and only if the matrix $B$ vanishes entirely. This
condition can be made intrinsic by observing that, if $\phi\colon \Lambda
\rightarrow H / \Lambda$ is the linear map encoding a tangent vector, then $B$
is a matrix representation for the linear map $\Lambda \cap P^{a_i} \to H /
(P^{a_i} + \Lambda)$ induced by $\phi$. Therefore it follows that, in the case
$\Lambda \in \Sigma_i^\circ$, $\phi$ described a tangent vector to $\Sigma_i$
if and only if $\phi(\Lambda \cap P^{a_i}) \subseteq (P^{a_i} +
\Lambda)/\Lambda$. This proves the claim, and the first statement of
the proposition.

The second statement follows by direct computation of the codimension
imposed by the conditions on the tangent space in the first part.
If we have $i \in S$, let $i_n$ denote the next (greater) element of $S$,
setting $i_n=r+1$ if $i$ is maximal in $S$. By starting from the condition
imposed at the maximal element of $S$, and inductively working downwards,
one computes that the codimension of the tangent space is given by
$$\sum_{i \in S} (i_n-i)(a_i-i).$$
Each term of this sum is always less than or equal to
$\sum_{j=i}^{i_n-1} (a_j-j)$, with equality if and only if there are no
actives indices strictly between $i$ and $i_n$. The proposition follows.
\end{proof}

\begin{cor}
\label{cor:tangent-schubert-circ} Given
$\Lambda \in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$,
there is a canonical isomorphism of vector spaces
$$T_{\Lambda}\Sigma_{P^{\bullet},a_{\bullet}}\cong
\left\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i})
\subseteq (P^{a_i}+\Lambda)/\Lambda \text{ for active } i=0,\ldots,r\right\}.$$
\end{cor}

Following Definition 4.1 of \cite{c-p3}, we define:

\begin{defn} \label{def:a-t} Two complete flags $P^{\bullet}$ and $Q^{\bullet}$ are called
\textbf{almost-transverse} if there exists an index $t\in\{1,\ldots,d-1\}$
such that
$$\dim P^i \cap Q^{d-i}
= \begin{cases} 0 & \text{ if $i\ne t$,}\\ 1 & \text{ if }i=t.\end{cases}$$
\end{defn}

More generally, we have the following statement, which is easy to check:

\begin{prop}\label{prop:sigma}
There is a unique permutation $\sigma \in S_d$ associated to the
flags $P^{\bullet}$ and $Q^{\bullet}$ with the property
that there exists a basis $e_1,\ldots,e_d$ for $H$ satisfying
$$e_i \in P^{i-1}\smallsetminus P^i, \quad \text{and} \quad
e_i \in Q^{\sigma(i)-1} \smallsetminus Q^{\sigma(i)}.$$

Such a basis can also be characterized by the property that for all indices
$i$ and $j$,
$P^i\cap Q^j$ is spanned by $\{e_1,\ldots,e_d\}\cap P^i\cap Q^j$.
In particular, if $\dim P^i\cap Q^j = 1$ then $P^i\cap Q^j$ contains one
of the $e_{\ell}$.
\end{prop}

\begin{defn} We refer to a basis as in Proposition \ref{prop:sigma} as
a $(P^{\bullet},Q^{\bullet})$-\textbf{basis}.
\end{defn}

Thus, following the notation of Proposition \ref{prop:sigma}, we have
that $P^{\bullet}=Q^{\bullet}$ if and only if $\sigma=\mathrm{id}$,
$P^{\bullet}$ and $Q^{\bullet}$ are transverse if and only if
$\sigma = \omega := (d,d-1,\ldots,1)$, and $P^{\bullet}$ and $Q^{\bullet}$
are almost-transverse if and only if $\sigma$ is the composition of $\omega$
with an adjacent transposition.

We set $\rho = \rho(1,r,d,a_{\bullet},b_{\bullet})$, so that
$$\rho-1=(r+1)(d-r-1)-\sum_{i=0}^r (a_i-i)-\sum_{i=0}^r (b_i-i)
= \rho(0,r,d-1,a_{\bullet},b_{\bullet})$$
is precisely the expected dimension of
$\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}}$.
Also recall the definition of the complete flags $\Lambda\cap P^\bullet$ and $\Lambda\cap Q^\bullet$ from Situation~\ref{sit:flags}.

\begin{thm}\label{thm:workhorse}
Given $\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$,
let $\sigma \in S_{r+1}$ denote the permutation associated to
$\Lambda\cap P^{\bullet}$ and $\Lambda\cap Q^{\bullet}$ in $\Lambda$
by Proposition \ref{prop:sigma}.
Given any $j\in \{0,\ldots,r\}$, let $$m(j) = \max\{a_i: i\text{ is active in $a_{\bullet}$ and $i\le j$}\},$$
setting $m(j) = 0$ if no such $a_i$ exists. Similarly, let
$$n(j) = \max\{b_i: i\text{ is active in $b_{\bullet}$ and $i\le \sigma(j)$}\},$$ or $n(j) = 0$ if no such $b_j$ exists.
Then
\begin{equation}\label{eq:the-answer}
\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
= \rho-1 + \sum_{j=0}^r \operatorname{codim}_H (P^{m(j)} + Q^{n(j)} + \Lambda).
\end{equation}
\end{thm}

\begin{proof}
Let $\lambda_0,\ldots,\lambda_r$ be a
$(\Lambda\cap P^{\bullet},\Lambda\cap Q^{\bullet})$-basis for
$\Lambda$.   Then for any $i$ active in $a_{\bullet}$, respectively $b_{\bullet}$, have
\begin{equation}\label{eq:lambda-basis}
\Lambda \cap P^{a_i} =\langle \lambda_i,\ldots,\lambda_r\rangle,\qquad \Lambda\cap Q^{b_i} = \langle \lambda_{\sigma^{-1}(i)}, \ldots,\lambda_{\sigma^{-1}(r)}\rangle.
\end{equation}
In other words, given any $j$, and any $i$ that is active in $a_{\bullet}$, we have $\lambda_j\in \Lambda\cap P^{a_i}$ if and only if $i\le j$.  Similarly, for any $i$ that is active in $b_{\bullet}$, we have $\lambda_j\in \Lambda\cap Q^{b_i}$ if and only if $i\le\sigma( j)$.
By Corollary~\ref{cor:tangent-schubert-circ}, we have isomorphisms
\begin{eqnarray*}
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
 &\cong& \{\phi\colon \Lambda\to H/\Lambda ~:~ \phi(\lambda_j) \in (P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda.\}\\
 &\cong&  \bigoplus_{j=0}^r \mathrm{Hom} \left(\langle \lambda_j\rangle, (P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda\right).\end{eqnarray*}
 We are thus reduced to computing the dimensions $(P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda$, which are equal to
$$\dim (P^{m(j)}+\Lambda)+ \dim (Q^{n(j)}+\Lambda)
- \dim (P^{m(j)}+ Q^{n(j)}+\Lambda)-\dim \Lambda.$$
 Moreover, the first two terms are determined by the fact that $\dim P^{m(j)}\cap \Lambda = r+1-m(j)$ and $\dim Q^{n(j)} \cap \Lambda = r+1-n(j)$, by assumption that $m(j)$ and $n(j)$ are active or are equal to $0$. A straightforward calculation produces~\eqref{eq:the-answer}.
\end{proof}

We observe that the well-known case of transverse flags follows
immediately from Theorem \ref{thm:workhorse}.

\begin{cor}\label{cor:transverse}
If $P^{\bullet}$ and $Q^{\bullet}$ are transverse, then
$\dim \, T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
= \rho-1 $ for all
$\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$.
\end{cor}

\begin{proof} Following the notation of the proof of Theorem
\ref{thm:workhorse}, for each $j$ we have
$\lambda_j  \in P^{m(j)} \cap Q^{n(j)} $ by construction.
Since $P^{\bullet}$ and $Q^{\bullet}$ are transverse, it follows that
$m(j) + n(j) < d$, so
$P^{m(j)} + Q^{n(j)} + \Lambda = P^{m(j)} + Q^{n(j)} = H$.
\end{proof}

More importantly, we can also deduce the desired statement in the
almost-transverse case.

\begin{cor}\label{cor:at}
Given $\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$,
suppose $P^{\bullet}$ and $Q^{\bullet}$ are almost-transverse, with $t+t'=d$
such that $\dim P^t \cap Q^{t'} =1$.

Suppose first that $t=a_i$ for $i$ active in $a_{\bullet}$, that $t'=b_{i'}$ for $i'$ active in $b_{\bullet}$, and that
$$P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}.$$
Then $$\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho.$$
If those conditions do not all hold, then $$\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho-1.$$
\end{cor}

\begin{proof}
Let $\lambda_0,\ldots,\lambda_r$ be a
$(\Lambda\cap P^{\bullet},\Lambda\cap Q^{\bullet})$-basis for
$\Lambda$.
First suppose that  $t=a_i$ for $i$ active in $a_{\bullet}$, and $t'=b_{i'}$ for $i'$ active in $b_{\bullet}$, and that $P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}.$  We will deduce that $\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho.$

We have that $\Lambda\cap P^t$ and $\Lambda\cap Q^{t'}$ are elements in the flags $\Lambda\cap P^\bullet$ and $\Lambda\cap Q^\bullet$ respectively with intersection $\Lambda\cap P^t \cap Q^{t'}$ of dimension 1.  Proposition~\ref{prop:sigma} implies that $P^t \cap Q^{t'} = \langle \lambda_j \rangle$ for a unique $j\in\{0,\ldots,r\}$.   By Theorem~\ref{thm:workhorse} it is enough to show that for each $j'\in\{0,\ldots,r\}$, $$\operatorname{codim}_H (P^{m(j')} + Q^{n(j')} + \Lambda) = \begin{cases}
1 &\text{ if }j'=j,\\
0&\text{ if }j'\ne j.
\end{cases}$$

Now, for $j'\ne j$, the fact that $\lambda_{j'}\in P^{m(j')}\cap Q^{n(j')}$ and $P^\bullet$ and $Q^\bullet$ are almost-transverse implies that either $m(j')+n(j')<d$, or that $m(j') = t$ and $n(j') = t'$.  But the latter case cannot be, since then both $\lambda_j, \lambda_{j'}\in P^{t}\cap Q^{t'}$, contradicting that $\dim P^{t}\cap Q^{t'}=1$.  Therefore $m(j')+n(j')<d$ and $$P^{m(j')}+ Q^{n(j')} = P^{m(j')}+ Q^{n(j')}+\Lambda = H,$$ as desired.

Next, to show that $\operatorname{codim}_H (P^{m(j)} + Q^{n(j)}+\Lambda )=1$, we claim that $m(j) = t$ and $n(j) = t'$. Recall that $a_i = t$ and $a_{i'} =t'$. By assumption, $i$ is active in $a_{\bullet}$ and $\lambda_j\in\Lambda\cap Q^{a_i}$, so $i\le j$ by ~\eqref{eq:lambda-basis}.   We want to show that $i$ is the largest active index in $a_{\bullet}$ with $i\le j$.  Indeed, if $l$ is active in $a_{\bullet}$ with $i<l\le j$, then $\lambda_j\in P^{a_l}\cap Q^{b_{i'}}$.    But now $a_l > a_i$, so $a_l + b_{i'} > a_i + b_{i'} =  t+t'=d$.  Therefore $P^{a_l}\cap Q^{b_{i'}}=0$, contradiction. A similar argument shows $n(j) = t'$.  Therefore,
$$\operatorname{codim}_H (P^{m(j)} + Q^{n(j)} + \Lambda) = \operatorname{codim}_H (P^{t} + Q^{t'} + \Lambda)=1,$$
since $P^{t} + Q^{t'} $  is a hyperplane in $H$, and $\Lambda$ is contained in it by assumption.

It remains to show that if the conditions in the statement of Corollary~\ref{cor:at} do not all hold, then $\dim T_\Lambda  (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})= \rho-1.$  We prove the contrapositive. Suppose that $\dim T_\Lambda  (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})>\rho-1.$  By Theorem~\ref{thm:workhorse}, there is an index $j$ such that $\operatorname{codim}_H (P^{m(j)} + Q^{n(j)} + \Lambda) >0$.  Again, given that $\lambda_j \in P^{m(j)} \cap Q^{n(j)}$ and that $P^\bullet$ and $Q^\bullet$ are almost-transverse, it follows that either $m(j)+n(j)<d$ or that $m(j)=t$ and $n(j)=t'$.  But $m(j)+n(j)<d$ would imply $P^{m(j)} + Q^{n(j)} = H$, contradicting the codimension statement. So $m(j)=t$ and $n(j)=t'$, implying that $t=a_i$ and $t'=b_{i'}$ for active indices $i$ and $i'$ in $a_{\bullet}$ and $b_{\bullet}$ respectively.
(It is not possible that $m(j) = 0$ or $n(j) = 0$, since $\operatorname{codim}_H P^{m(j)}+Q^{n(j)}+\Lambda>0$.)
Furthermore,
$$\langle \lambda_j \rangle = P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}$$
where the last containment holds again by the codimension assumption.

Summarizing, we have shown that the {\em only} way that $\dim T_\Lambda  (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})> \rho-1$ is for all the conditions in the statement of Corollary~\ref{cor:at} to hold, in which case we have already proved that the dimension is exactly $\rho$.
\end{proof}

\begin{rem}\label{rem:arbitrary-flags}

For arbitrary flags $P^{\bullet}$ and $Q^{\bullet}$ and $\Lambda\in\Sigma^\circ_{P^\bullet,a_{\bullet}}\cap \Sigma^\circ_{Q^\bullet,b_{\bullet}}$, let $\tau\in S_d$ be
the associated permutation from Proposition
\ref{prop:sigma} (maintaining other notation as in Theorem
\ref{thm:workhorse}). Then the extent to which
the dimension of the tangent space at $\Lambda$ of
$\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}}$
exceeds $\rho-1$ can be bounded in terms of $\tau$ as follows. We have:
\begin{equation} \label{eq:coxeter}
\dim T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
 \leq (\rho-1) + \operatorname{inv}( \omega \tau ).
\end{equation}
Here $\omega$ denotes the decreasing permutation $(d,d-1,\cdots,1)$, and $\operatorname{inv}(\omega\tau)$
denotes the inversion number of $\omega\tau$, i.e.\ the number of $i<j$ with
$\omega\tau(i)>\omega\tau(j)$.\footnote{If one views $S_d$ as a Coxeter group
with reflections being the adjacent transpositions, then $\operatorname{inv}(\omega \tau)$
is also the Coxeter length of $\omega \tau$.}
We briefly sketch a proof of this more general inequality.

Using the second part of Proposition \ref{prop:sigma}, it follows that for
each $i,j$, we have
$$ \dim P^i \cap Q^j = \# \{ i' \geq i:\ \tau(i') \geq j \}. $$
From this it follows that $\dim P^i \cap Q^j > \dim P^{i+1} \cap Q^j$ if and
only if $\tau(i) \geq j$. Now, since we have
$P^{a_i} \cap Q^{b_{\sigma(i)}} \neq P^{a_i+1} \cap
Q^{b_{\sigma(i)}}$, we find
that $b_{\sigma(i)} \leq \tau(a_i)$. Note also that for all $j$, $m(j) \leq a_j$ and $n(j) \leq b_{\sigma(j)}$. Then:

\begin{eqnarray*}
\sum_{j=0}^r \operatorname{codim} ( \Lambda + P^{m(j)} + Q^{n(j)} ) &\leq&
\sum_{j=0}^r \operatorname{codim}(P^{m(j)} + Q^{n(j)})\\
&\leq& \sum_{j=0}^r \operatorname{codim}(P^{a_j} + Q^{b_{\sigma(j)}})\\
&\leq& \sum_{j=0}^r \operatorname{codim}( P^{a_j} + Q^{\tau(a_j)} )\\
&\leq& \sum_{j=0}^{d-1} \operatorname{codim}( P^j + Q^{\tau(j)}).
\end{eqnarray*}

Using that
$\dim(P^j \cap Q^{\tau(j)})=\#\{j' \geq j: \tau(j') \geq \tau(j)\}$, we
compute that $\displaystyle \sum_{j=0}^{d-1} \operatorname{codim}( P^j
+ Q^{\tau(j)}) = \operatorname{inv}(\omega \tau)$, and the inequality \eqref{eq:coxeter}
follows from \eqref{eq:the-answer}.
When $P^{\bullet}$ and $Q^{\bullet}$ are almost-transverse, $\operatorname{inv}(\omega
\tau) = 1$, and
the precise statement in Corollary
\ref{cor:at} can be deduced from characterizing the equality cases of the four inequalities above in the case where $\omega \tau$ is equal to an
adjacent transposition.
\end{rem}

\section{Linear series in positive genera}\label{sec:proofs}

We begin with a proposition that will show that our smoothness result, Theorem~\ref{thm:main} to be proved below, is sharp.

\begin{prop}\label{prop:non-smooth}
In the situation of Theorem \ref{thm:bg},
every point of
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ in the complement of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is singular.
\end{prop}

\begin{proof} This is a consequence of the standard construction of
the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$: we let
$\wt{{\mathscr L}}$ be a Poincar\'e line bundle on $X \times \operatorname{Pic}^d(X)$.
Take a sufficiently ample\footnote{Precisely, of degree strictly greater
than $2g-2$} effective divisor $D$ on $X$ with support
disjoint from $P$ and $Q$, and write $\wt{D}=D \times \operatorname{Pic}^d(X)$. Writing $p\colon X\times \operatorname{Pic}^d(X)\to\operatorname{Pic}^d(X)$ for projection, let
$G$ be the relative Grassmannian
$\operatorname{Gr}(r+1,p_{*} \wt{{\mathscr L}}(\wt{D}))$, equipped with structure map $\pi\colon G\to \operatorname{Pic}^d(X)$.  Let $\wt{{\mathscr V}} \hookrightarrow \pi^* p_{*} \wt{{\mathscr L}}(\wt{D})$ denote the universal subbundle.
Then $G^r_d(X)$ is cut out in $G$ by the condition that the induced map
$$\wt{{\mathscr V}} \hookrightarrow \pi^*p_{*}\left( \wt{{\mathscr L}}(\wt{D})|_{\wt{D}}\right)$$
vanishes identically. Because we have chosen $D$ to have support disjoint
from $P$ and to be sufficiently ample, the space
$G^r_d(X,(P,a_{\bullet}))$ is cut out by imposing the
additional Schubert condition that the maps
$$\wt{{\mathscr V}} \hookrightarrow \pi^* p_{*}\left( \wt{{\mathscr L}}(\wt{D})|_{a_j P}\right)$$
have rank at most $j$ for each $j$.
Imposing the analogous condition at
$Q$, we obtain $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ as an
intersection of three conditions: a determinantal condition (in fact a
complete intersection), and two relative Schubert cycles. It is routine
to check that for
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ to have dimension
$\rho(g,r,d,a_{\bullet},b_{\bullet})$, as asserted by Theorem \ref{thm:bg},
these three
conditions must intersect in the maximal codimension. Given that we know
from Theorem \ref{thm:bg} that
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ does in fact have dimension
$\rho(g,r,d,a_{\bullet},b_{\bullet})$, it then further
follows that in order for
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ to be smooth at any point,
that point must lie in the smooth locus of each of the three conditions,
and in particular of the two Schubert cycles.

But we claim that
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ consists precisely of the
points of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ which lie in the
smooth locus of both relative Schubert cycles.
Indeed, each relative Schubert cycle is nothing but a locally constant family of Schubert varieties over the base $G$, so we are done by the standard characterization of the smooth locus of a Schubert variety (see the second part of Proposition \ref{prop:tangent-schubert}).
\end{proof}

We now use our calculations in Grassmannians in \S2 to complete the proof of our
main theorem, beginning with the case of genus $1$ in
Theorem~\ref{thm:genus-1}.

\begin{proof}[Proof of Theorem \ref{thm:genus-1}]
Set $\rho=\rho(1,r,d,a_{\bullet},b_{\bullet})$.
We may assume $d>0$, as otherwise the result is trivial. Thus,
$G^r_d(X)$ is a Grassmannian bundle over $\operatorname{Pic}^d(X)$, with the fiber
over a line bundle ${\mathscr L}$ being canonically identified with
$\operatorname{Gr}(r+1,\Gamma(X,{\mathscr L}))\cong \operatorname{Gr}(r+1,d)$. The condition imposed by
requiring vanishing sequence at least $a_{\bullet}$ at $P$ then gives
a Schubert cycle in each fiber, corresponding to the complete flag determined by
vanishing order at $P$. The codimension of spaces in the flag corresponds
precisely to vanishing order except over the point ${\mathscr L} \cong {\mathscr O}_X(dP)$,
where no sections vanish to order precisely $d-1$, and vanishing to order
$d$ imposes codimension only $d-1$. Consequently, there are
two possibilities for $G^r_d(X,(P,a_{\bullet}))$. First, if $a_r<d$, it is
a relative Schubert cycle of codimension $\sum_j (a_j-j)$ in
$G^r_d(X)$, Cohen-Macaulay and flat over $\operatorname{Pic}^d(X)$.
Or,
if $a_r=d$, it is supported entirely over ${\mathscr L} \cong {\mathscr O}_X(dP)$ (even
scheme-theoretically), and is still a Schubert cycle, but of codimension
$(\sum_j (a_j-j))-1$. The same analysis applies to
$G^r_d(X,(Q,b_{\bullet}))$, so we find that every fiber of
\begin{equation}\label{eq:str-map}
G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \operatorname{Pic}^d(X)
\end{equation}
is an intersection of a pair of Schubert cycles. The basic properties of
the map \eqref{eq:str-map} are analyzed for instance in Lemma 2.1 of
\cite{os18} and Proposition 2.1 of \cite{os26}; we review the main points
of this analysis in order to carry out the necessary tangent space analysis.

First, we see that in most fibers of \eqref{eq:str-map}, the relevant
Schubert cycles are associated to transverse flags:
the only way in which the flags fail to be transverse is if
${\mathscr L} \cong {\mathscr O}_X(aP+(d-a)Q)$ for some $a\in\{1,\ldots,d-1\}$, which is
unique by genericity of $P$ and $Q$; then the conditions of
vanishing to order $a$ at $P$ and $d-a$ at $Q$ intersect in dimension $1$
instead of dimension $0$. Thus, on fibers of \eqref{eq:str-map}
over points not of the form
${\mathscr O}_X(aP+(d-a)Q)$ for $0 \leq a \leq d$, we have that the Schubert indexing
matches vanishing sequences, and the flags are transverse, so
the standard theory (see for instance Corollary \ref{cor:transverse})
gives us that (on these fibers) the space
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth (of
relative dimension $(r+1)(d-r-1)-\sum_j (a_j-j)-\sum_j (b_j-j) = \rho-1$) over
$\operatorname{Pic}^d(X)$, and hence smooth of relative dimension $\rho$
over $\operatorname{Spec} k$. Similarly, it is easily
verified that we still obtain Richardson varieties over ${\mathscr O}_X(aP+(d-a)Q)$
for $0<a<d$
unless $a$ occurs in $a_{\bullet}$ and $d-a$ occurs in $b_{\bullet}$.
Thus, we obtain the desired statement in these cases.
On the other hand, if $a=0$ or $a=d$, we have transverse intersection of
flags, but a potential difference in indexing. In the case $a=d$, the
difference in indexing arises only in that imposing vanishing order $d$ at
$P$ is a codimension $d-1$ condition. Thus, this can only affect the final
term in the vanishing sequence, which is irrelevant for determining
membership in
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$. We conclude
that---whether or not $a_r=d$---the fiber of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ over
${\mathscr O}_X(dP)$ precisely corresponds to the open subset addressed
in Corollary \ref{cor:transverse}, and hence has the desired smoothness
property. The case that $a=0$ is the same, with $Q$ in place of $P$.

It thus remains to analyze the fibers with ${\mathscr L}={\mathscr O}_X(aP+(d-a)Q)$, for
$0<a<d$, and with $a$
occurring in $a_{\bullet}$ and $d-a$ occurring in $b_{\bullet}$.
Our hypothesis on $P-Q$ implies that
${\mathscr O}_X(aP+(d-a)Q) \not \cong {\mathscr O}_X(a'P+(d-a')Q)$ for any $a \neq a'$,
so even in these cases, our flags in $\Gamma(X,{\mathscr L})$ are almost-transverse.
In the case that $a=a_j$ and $d-a=b_{r-j}$ for some $j$, then
$G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is supported
(scheme-theoretically) in the given fiber, and one checks (see
the proof of Proposition 2.1 of \cite{os26}) that the nonempty fiber can still
be described as a Richardson variety, by replacing $a_j$ with $a-1$, and
changing the choice of codimension-$a$ subspace in the first flag. Because
of this modification to the flag, we will still have that
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ corresponds precisely to
the open subset treated in Corollary \ref{cor:transverse}, and is hence
smooth (but this time of dimension $\rho$).

Finally, we consider the case that $a_j+b_{r-j}<d$ for all $j$, but we
have ${\mathscr L}={\mathscr O}_X(aP+(d-a)Q)$, and $a=a_j$ and $d-a=b_{j'}$ for some $j,j'$
with $j+j'>r$; in particular, we must have $0<a<d$. See Example~\ref{ex:0202} for what is essentially the smallest nontrivial example of this case.
In this situation,
the given fiber of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ over $\operatorname{Pic}^d(X)$ may
be singular or even reducible, but at least it is pure of dimension
$\rho-1$: see the proof of Proposition 2.1 of \cite{os26}.
Moreover, as we have observed, the fiber is an intersection
of Schubert cycles associated to almost-transverse flags, so we can
invoke Corollary \ref{cor:at} to conclude that the fiber of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ has tangent space
dimension equal to $\rho-1$ or $\rho$ everywhere. Moreover, the latter
occurs precisely at linear series $({\mathscr L},V)$ satisfying the following
conditions:
\begin{enumerate}
\item $V$ contains
a section $s$ vanishing to order $a$ at $P$ and $d-a$ at $Q$;
\item $V$ is
contained in the linear span of the spaces of sections vanishing to order $a$ at
$P$ and order $d-a$ at $Q$;
\item there is some $j$ with $a=a_j$ and $j$ active
in $a_{\bullet}$ in the sense of Definition \ref{defn:active};
\item there is some
$j'$ with $d-a=b_{j'}$ and $j'$ active in $b_{\bullet}$.
\end{enumerate}

Thus, in order to complete the proof of the
theorem, we will prove that the space
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth of dimension
$\rho$ at every point of the given fiber by showing that
if $({\mathscr L},V)$ is a point at which the tangent space of the fiber has
dimension $\rho$, then every tangent vector of the total space at $({\mathscr L},V)$
is in fact vertical.
Accordingly, given $({\mathscr L},V)$ satisfying the four conditions above, suppose $(\wt{{\mathscr L}},\wt{V})$ is a first-order deformation of
$({\mathscr L},V)$, and let $s \in V$ be a section vanishing to order $a$ at $P$ and
$d-a$ at $Q$.  We claim that $s$ has a lift $\tilde{s} \in \wt{V}$ which vanishes
(scheme-theoretically) to order $a$ at $P$.
Indeed, in the notation of the proof of Proposition~\ref{prop:non-smooth}, recall that on $G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ we have a map of vector bundles
\begin{equation}\label{eq:rank-at-most-j}
\Phi\colon\tilde{{\mathscr V}}\hookrightarrow\pi^*p_*\left(\tilde{{\mathscr L}}(\tilde{D})|_{a_jP}\right)
\end{equation}
which has rank at most $j$.  The assumption that $({\mathscr L},V)$ lies
in the open subset $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$,
together with the
fact that $j$ is active in $a_{\bullet}$, says that on every point in an open neighborhood of $({\mathscr L},V)$, the map~\eqref{eq:rank-at-most-j} has rank {\em exactly} $j$. In this situation, a standard argument shows that $\operatorname{ker}\Phi$ is locally free, and that the restriction map $(\operatorname{ker}\Phi)|_{(\tilde{{\mathscr L}},\tilde{V})} \rightarrow (\operatorname{ker}\Phi)|_{({\mathscr L},V)}$ is surjective. A sketch of this standard argument is as follows. The cokernel of $\Phi$ must be locally free (see e.g.~\cite[\S16.7]{ei1}), and hence also the image and kernel. Therefore, the short exact sequences $0\to\operatorname{ker}\Phi\to \tilde{{\mathscr V}} \to \operatorname{im}\Phi\to0$ and $0\to\mathrm{im}\Phi\to\pi^*p_*\left(\tilde{{\mathscr L}}(\tilde{D})|_{a_jP}\right) \to\operatorname{cok}\Phi\to 0$ remain exact after base change, and hence taking kernels commutes with base change.

From surjectivity of the restriction map above, we deduce that our section $s$ admits a lift $\tilde{s}$ that also vanishes (scheme-theoretically) to order $a$ at $P$.
Similarly, $s$ must have
another lift which vanishes
(scheme-theoretically) to order $d-a$ at $Q$; since it is another lift of $s$, it can be
expressed as $\tilde{s}+\epsilon v$ for some $v \in V$.

Now, recall our hypothesis that $V$ is contained in the
span of sections vanishing to order at least $a$ at $P$ and at least $d-a$ at $Q$.
Write $v=v_1 + v_2$, where $v_1$ vanishes to order at least $a$ at
$P$ and $v_2$ vanishes to order at least $d-a$ at $Q$. But then
$\tilde{s}+ \epsilon v_1$ still vanishes to order $d-a$ at $Q$, and also vanishes to
order $a$ at $P$. This forces $\wt{{\mathscr L}}$ to be the trivial deformation of ${\mathscr L}$,
yielding the desired verticality
assertion and the theorem.
\end{proof}

\begin{rem}\label{rem:reduced-fibers} Although it was shown in
\cite{os26} that $G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$
is reduced, we are not aware of a proof in the literature that every fiber of
$G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \operatorname{Pic}^d(X)$
is reduced, even for genus $1$. However, this genus-$1$ case follows from
the proof of Theorem \ref{thm:genus-1}.
Indeed, the fibers can be expressed as intersections of a pair of Schubert
varieties having the expected dimension, and they are therefore
Cohen-Macaulay. Furthermore, our proof produces dense open subsets of
each fiber which are smooth, so we conclude reducedness.
\end{rem}

To conclude the proof of our main theorem, we need to make use of the
Eisenbud-Harris theory of limit linear series. We first set up notation
for our reducible curves, and recall the relevant definitions.

\begin{sit}\label{sit:chain} Fix $g,d,n$. Let $Z_1,\dots,Z_n$
be smooth projective curves, with (distinct) points $P_i, Q_i$ on $Z_i$ for each
$i$, and let $X_0$ be the nodal curve obtained by gluing $Q_i$ to $P_{i+1}$
for $i=1,\dots,n-1$.
\end{sit}

\begin{defn}\label{defn:lls} Given $r,d$ a \textbf{limit linear series}
of dimension $r$ and degree $d$ on $X_0$ consists of a tuple $({\mathscr L}^i,V^i)$ of
linear series of dimension $r$ and degree $d$ on the $Z_i$, satisfying the
following condition: if $a^i_{\bullet},b^i_{\bullet}$ are the vanishing
sequences of $({\mathscr L}^i,V^i)$ at $P_i$ and $Q_i$ respectively, then we require
\begin{equation}\label{eq:eh-ineq}
b^i_j+a^{i+1}_{r-j} \geq d
\end{equation}
for all $i=1,\dots,n-1$ and $j=0,\dots,r$. If \eqref{eq:eh-ineq} is an
equality for all $i,j$, we say that the limit linear series is
\textbf{refined}.

The space of all such limit linear series on $X_0$ is denoted by
$G^r_d(X_0)$. If we have sequences $a_{\bullet}$ and $b_{\bullet}$,
we also have the closed subscheme
$G^r_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet})) \subseteq G^r_d(X_0)$
consisting of limit linear series such that, following the above notation,
we have $a^1_{\bullet} \geq a_{\bullet}$ and $b^n_{\bullet} \geq b_{\bullet}$.
Finally, denote by
$G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet})) \subseteq
G^r_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ the open subscheme
consisting of refined limit linear series which further satisfy
\begin{eqnarray*}
\dim V^1(-a_j P_1) =r+1-j &\text{for $j>0$ active in $a$,}\\
\dim V^n(-b_j Q_n) =r+1-j &\text{for $j>0$ active in $b$.}
\end{eqnarray*}
We comment that these last two conditions can be re-expressed purely in terms of $a^1_\bullet$ and $b^n_\bullet$ as follows: for all $j>0$ active in $a$, we require
$\#\{j':a^1_{j'} \ge a_j\} = r+1-j$, and similarly for $b$.
\end{defn}

We are now ready to prove our main smoothness result.

\begin{proof}[Proof of Theorem \ref{thm:main}]
In Situation \ref{sit:chain}, observe that we can decompose the limit
linear series space
$G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ into disjoint open
subsets according to the vanishing sequences at each node, i.e.~according
to the possible values in the left hand side of the equalities~\eqref{eq:eh-ineq}.
Then each such open subset is almost a product over $i$ of spaces of the form
$G^{r,\circ}_d(Z_i,(P_i,a^i_{\bullet}),(Q_i,b^i_{\bullet}))$. In fact, it
is an open subset of this product, since the refinedness condition completely
fixes the vanishing sequences at the nodes. If further
each $Z_i$ has genus $0$ or $1$, and for the $Z_i$ of genus $1$ we
suppose that $P_i-Q_i$ is not $m$-torsion for any $m \leq d$, then we know
that each $G^{r,\circ}_d(Z_i,(P_i,a^i_{\bullet}),(Q_i,b^i_{\bullet}))$ is
smooth. Indeed, the genus-$1$ case is Theorem \ref{thm:genus-1}, while the
genus-$0$ case is well known, but follows in particular immediately from
Corollary \ref{cor:transverse} taking into account that
$\dim \Gamma({\mathbb P}^1,{\mathscr O}(d))=d+1$, so there is a shift of $1$ in the value
of $d$. We thus conclude that
$G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ is also smooth,
of dimension $\rho(g,r,d,a_{\bullet},b_{\bullet})$.

Now, fix $n=g$ and suppose each $Z_i$ has genus $1$.
Let $B$ be the spectrum of a discrete valuation ring, and $\pi:X \to B$ be
a flat, proper family family of curves of genus $g$, with
$X$ regular, the generic fiber $X_{\eta}$ smooth, and the special fiber
isomorphic to $X_0$. Further assume that $\pi$ has sections $P$, $Q$,
specializing to $P_1$ and $Q_n$ respectively on $X_0$.

Suppose that we have a closed point of
$G^{r,\circ}_d(X_{\eta},(P_{\eta},a_{\bullet}),(Q_{\eta},b_{\bullet}))$.
Extend the base so that the corresponding linear series is defined
on $X_{\eta}$, and then extend further so that all ramification points are also
rational over the base field. Blow up the nodes in $X_0$ as necessary
to resolve any resulting singularities,\footnote{The preceding constitutes
an alternative for the argument of Theorem 2.6 of \cite{e-h1}, avoiding
invocation of the stable reduction theorem.} and finally, blow up $P_1$ and
$Q_n$ as necessary so that no generic ramification point distinct from
$P$ or $Q$ limits to $P_1$ or $Q_n$ in the special fiber.
Denote the
resulting family by $\pi':X' \to B'$, and the special fiber by $X'_0$,
and write $P'$ and $Q'$ (respectively, $P'_1$ and $Q'_n$) for the
resulting sections of $\pi'$ and their restrictions to $X'_0$.
Then $X'_0$ is obtained by $X_0$ by base extension and insertion of
chains of genus-$0$ curves at the nodes and at $P_1$ and $Q_n$.
By construction, none of the ramification points on $X_\eta$ can specialize to nodes of
$X'_0$, so by Proposition 2.5 of \cite{e-h1} (and using the characteristic
$0$ hypothesis), the extension of the given linear series is a refined limit
linear series on $X'_0$. Moreover, by the same argument, the ramification
at $P'_1$ and at $Q'_n$ must be precisely equal to the ramification at
$P_{\eta}$ and $Q_{\eta}$, so
that the induced limit linear series lies in
$G^{r,\circ}_d(X'_0,(P'_1,a_{\bullet}),(Q'_n,b_{\bullet}))$.
But as we have discussed above, this space is smooth. Moreover,
by \cite{o-m1} (see also Theorem 3.4 of \cite{os26} for the situation
with imposed ramification) there is a flat relative moduli space recovering
linear series on the generic fiber and limit linear series on the special
fiber.\footnote{In fact, since we only need refined limit linear
series for our specialization argument, it is likely possible to make a flatness argument
using only the original Eisenbud-Harris construction of \cite{e-h1}, rather than appealing to the general results of \cite{o-m1,os26}.  But we are not aware of a reference for the more restrictive statement.}
It follows that the original point of
$G^{r,\circ}_d(X_{\eta},(P_{\eta},a_{\bullet}),(Q_{\eta},b_{\bullet}))$
must have been smooth as well.

Now, since the spaces we are considering are in general not proper, the condition that $G^{r,\circ}_d$ is smooth is not open in families. However, the condition does define a constructible subset of ${\mathcal M}_{g,2}$,
and the generic fibers of the possible families $\pi$ as above correspond to
a Zariski-dense subset, so we conclude the main smoothness statement of the
theorem. The fact that the remaining points are not smooth is Proposition
\ref{prop:non-smooth}.

The statement on codimension of singularities follows
from the observation that a point in the complement of
$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$
is a union of closed subvarieties of the form $G^{r}_d(X,(P,a'_{\bullet}),(Q,b'_{\bullet}))$, where $a'_\bullet \ge a$ and $b'_\bullet \ge b$ are sequences such that for some $j>0$ active in $a_\bullet$, we have $a'_{j-1} \ge a_j$, implying that $a'_{j-1}\ge a_{j-1}+2$ and $a_j' \ge a_j+1$ and hence $\sum a'_\bullet \ge \sum a_\bullet+3$; or analogously for $b_\bullet$.
We then conclude normality from the Cohen-Macaulayness and
Serre's criterion, and the irreducibility statement follows immediately
from the connectedness in the case $\widehat{\rho} \geq 1$.
\end{proof}

\begin{ex}\label{ex:0202}
We provide here essentially the smallest interesting example in the $g=1$ case, exhibiting a fiber of $G^r_d(X, (P, a_\bullet), (Q,b_\bullet))\to \operatorname{Pic}^d(X)$ that is not a Richardson variety, and is in fact reducible.  Let $a_\bullet = b_\bullet = (0,2)$, let ${\mathscr L} = {\mathscr O}_X(2P + 2Q)$, and consider the fiber of $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ over ${\mathscr L}$.  This fiber has two irreducible components $Z_1$ and $Z_2$, each isomorphic to $\mathbb{P}^2$, meeting along a $\mathbb{P}^1$. It may be described as the variety of lines in $\mathbb{P}^3$ that meet two fixed lines that themselves intersect at a point.

In this situation, the vertical tangent space at a point in $Z_1\cap Z_2\cong \mathbb{P}^1$ has dimension jumping up to $3$.  Now, the fiber of $G^{1,\circ}_4(X, (P, a_\bullet), (Q,b_\bullet))$ over ${\mathscr L}$ is obtained from that of $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ by removing two points of $Z_1\cap Z_2$. Those two points correspond to the space of sections of ${\mathscr L}$ vanishing to order at least $2$ at $P$, respectively the space of sections of ${\mathscr L}$ vanishing to order at least $2$ at $Q$.
Then Theorem~\ref{thm:genus-1} asserts that on $Z_1\cap Z_2$, except for at those two points, $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ has no horizontal tangent vectors.
\end{ex}

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


girl – "Your Dying Charlotte"
Lavada May Dermott, postmortem albumen print on cardboard, 1904. Ann Longmore-Etheridge Collection.
Lavada May Dermott, seen above laid out in her parents' home, surrounded by funeral flowers, was born 9 May, 1888, in Goshen, Belmont County, Ohio, to Charles Evans Dermott (1849-1907) and his wife, Sarah Jane Stewart (1853-1936). Sarah was the daughter of Eleazar Evan Stewart (1834-1910) and Honor Brown (1835-1914).
Lavada's parents married in Goshen 26 January, 1871. They first make an appearance as a family unit on the 1880 Census of that township. Evans Dermott—he went by his middle name—listed his occupation as "huxter." His father, Irish-born farmer William Dermott (1811-1896) was also a part of the household—his wife, Eliza Kelly (1813-1879), having died the year before.
Evans and Sarah had together four children: William Wilber (1871-1947), Charles E. Dermott (19 June, 1881-23 September, 1944), Lavada, and Lillie F. Dermott (1895-1987). On 20 May, 1900, the eldest son, William, married Grace Stanley King. William spelled the family name as Der Mott and is buried thusly in Forest Rose Cemetery, Lancaster, Ohio. By 1910, the Der Motts lived in Cleveland, Ohio. William became a garment cutter for a clothing company and eventually was the manager of a clothing store. He worked in that position as late as 1940. They had two children: Neil (b. 1902) and William P. (b. 1903).


text:
Q: how do i generate a heat map of a detected object based on its position using opencv python I need to generate the heat map of a object based on its position
example: the detected green ball in a video frame . If it stays in a position for a long duration then that place should be red and the positions in the frame where the ball has passes for a short duration must be in blue in that way i need to generate a heat map . Thanks in advance

A: Well what you can do here is
1) First define a heatmap as the size of the image
  heatmap = np.zeros_like(img[:,:,0]).astype(np.float)

2) Since you already have the detected object and its position, in the    heatmap, add 1 to all pixels in the bounding box of the object.
heatmap[box[0][1]:box[1][1], box[0][0]:box[1][0]] += 1

3) you can than apply thresholding by setting all pixels which are below a threshold value in the heatmap to be 0. You can choose the threshold value to be 1 so that the region inside your bounding box remains in the heatmap
heatmap[heatmap <= threshold] = 0


Spring and it's time to harvest a little Rhubarb on our allotment, did you know that it's a really easy vegetable to grow.
It can be planted into the garden or you can plant it into a big tub which could sit out onto your yard.
We have a lovely recipe that we know you are going to love and it's a great for all the family to join in , make and of course eat!
Method - set the oven to 180°c/gas 4. Line a tray with 12 cupcake cases.
Beat butter and caster sugar together until light. Add the egg followed by the flour. Beat well for 2-3 mins or until smooth.
Fill each case with a tbsp of mixture.
Bake for 15-20 mins until golden. Cool cakes completely.
Make the jam by combining rhubarb, lemon and zest in a pan and bring to the boil .Reduce heat and simmer for 10mins or so. Add sugar and simmer for 10-15 mins, leave to cool.
Once the jam is cool, whip the cream and icing sugar together then spoon in 2 tbsp of jam and mix in.
Cut the top off the cooled cake and cut in two to make the wings.
Spoon a teaspoon of jam into the top of the cupcake then pipe a swirl of cream on top.
Place the wings on to resemble a butterfly.


Tonight on the mid-season finale of Supernatural titled, 'The Spear', Fan Favorite 'Garth' returns as Sam and Dean ask him for help to find out just what Michael is planning! Check out the promo below and share your thoughts AFTER THE JUMP. Supernatural airs Tonight at 7pm Central on The CW!


text:
#include "scriptable_options.h"
#include "options_interface.h"
#include "small_object.h"

namespace ggadget {

class ScriptableOptions::Impl : public SmallObject<> {
 public:
  Impl(OptionsInterface *options, bool raw_objects)
      : options_(options), raw_objects_(raw_objects) {
  }

  void Add(const char *name, const JSONString &value) {
    options_->Add(name, Variant(value));
  }

  // If an item doesn't exist, returns a blank string "" (JSON '""').
  JSONString GetDefaultValue(const char *name) {
    Variant value = options_->GetDefaultValue(name);
    return value.type() == Variant::TYPE_JSON ?
           VariantValue<JSONString>()(value) : JSONString("\"\"");
  }

  void PutDefaultValue(const char *name, const JSONString &value) {
    options_->PutDefaultValue(name, Variant(value));
  }

  // If an item doesn't exist, returns a blank string "" (JSON '""').
  JSONString GetValue(const char *name) {
    Variant value = options_->GetValue(name);
    return value.type() == Variant::TYPE_JSON ?
           VariantValue<JSONString>()(value) : JSONString("\"\"");
  }

  void PutValue(const char *name, const JSONString &value) {
    options_->PutValue(name, Variant(value));
  }

  // If an item doesn't exist, returns undefined (JSON "").
  JSONString OldGetDefaultValue(const char *name) {
    Variant value = options_->GetDefaultValue(name);
    return value.type() == Variant::TYPE_JSON ?
           VariantValue<JSONString>()(value) : JSONString("");
  }

  // If an item doesn't exist, returns undefined (JSON "").
  JSONString OldGetValue(const char *name) {
    Variant value = options_->GetValue(name);
    return value.type() == Variant::TYPE_JSON ?
           VariantValue<JSONString>()(value) : JSONString("");
  }

  OptionsInterface *options_;
  bool raw_objects_;
};

ScriptableOptions::ScriptableOptions(OptionsInterface *options,
                                     bool raw_objects)
    : impl_(new Impl(options, raw_objects)) {
}

void ScriptableOptions::DoRegister() {
  OptionsInterface *options = impl_->options_;
  RegisterProperty("count",
                   NewSlot(options, &OptionsInterface::GetCount), NULL);
  RegisterMethod("exists", NewSlot(options, &OptionsInterface::Exists));
  RegisterMethod("remove", NewSlot(options, &OptionsInterface::Remove));
  RegisterMethod("removeAll", NewSlot(options, &OptionsInterface::RemoveAll));
  RegisterMethod("encryptValue",
                 NewSlot(options, &OptionsInterface::EncryptValue));

  if (impl_->raw_objects_) {
    // Partly support the deprecated "item" property.
    RegisterMethod("item", NewSlot(options, &OptionsInterface::GetValue));
    // Partly support the deprecated "defaultValue" property.
    RegisterMethod("defaultValue",
                   NewSlot(options, &OptionsInterface::GetDefaultValue));
    // In raw objects mode, we don't support the deprecated properties.
    RegisterMethod("add", NewSlot(options, &OptionsInterface::Add));
    RegisterMethod("getDefaultValue",
                   NewSlot(options, &OptionsInterface::GetDefaultValue));
    RegisterMethod("getValue", NewSlot(options, &OptionsInterface::GetValue));
    RegisterMethod("putDefaultValue",
                   NewSlot(options, &OptionsInterface::PutDefaultValue));
    RegisterMethod("putValue", NewSlot(options, &OptionsInterface::PutValue));

    // Register the "default" method, allowing this object be called directly
    // as a function.
    RegisterMethod("", NewSlot(options, &OptionsInterface::GetValue));
  } else {
    // Partly support the deprecated "item" property.
    RegisterMethod("item", NewSlot(impl_, &Impl::OldGetValue));
    // Partly support the deprecated "defaultValue" property.
    RegisterMethod("defaultValue",
                   NewSlot(impl_, &Impl::OldGetDefaultValue));
    RegisterMethod("add", NewSlot(impl_, &Impl::Add));
    RegisterMethod("getDefaultValue",
                   NewSlot(impl_, &Impl::GetDefaultValue));
    RegisterMethod("getValue", NewSlot(impl_, &Impl::GetValue));
    RegisterMethod("putDefaultValue",
                   NewSlot(impl_, &Impl::PutDefaultValue));
    RegisterMethod("putValue", NewSlot(impl_, &Impl::PutValue));

    // Register the "default" method, allowing this object be called directly
    // as a function.
    RegisterMethod("", NewSlot(impl_, &Impl::OldGetValue));

    // Disable the following for now, because it's not in the public API.
    // SetDynamicPropertyHandler(NewSlot(impl_, &Impl::GetValue),
    //                           NewSlot(impl_, &Impl::PutValue));
  }
}

ScriptableOptions::~ScriptableOptions() {
  delete impl_;
}

const OptionsInterface *ScriptableOptions::GetOptions() const {
  return impl_->options_;
}

OptionsInterface *ScriptableOptions::GetOptions() {
  return impl_->options_;
}

} // namespace ggadget

redpajama_set_name: RedPajamaGithub