Hello, everyone. So there's been some breaking news and I'm not super prepared for this video, but I thought it was very important that I got this out as soon as possible. So I'm just gonna be following the facts that I see posted online. I recently made a video on the DXC case with Wayne Shung and Paul Piglesheimer. They're actually charged with burglary and theft for rescuing sick piglets from a huge factory farm owned by Smithfield in Utah. They were actually acquitted, which was amazing news because it sets the precedent that rescuing animals is not a crime. Unfortunately, we can't say the same for the next case. This case takes place in Canada and it represents a group of activists dubbed the Excelsior 4. I'm going to read out the details of the case. This is an absolute injustice. The Excelsior 4 were charged after exposing criminal activity at Excelsior Hog Farm in Abbotsford, Canada. Animal activists Amy Serrano, Nick Schaefer, Roy Susano, and Jeff Regia, known as the Excelsior 4, faced 21 indelible offenses, the highest criminal level in Canadian court, for revealing what happens to animals inside factory farms. In the spring of 2019, video footage taken from inside Excelsior Hog Farm and released by Peter showed crowded pens full of thousands of pigs, suffering from hernias, bloody lacerations, and golf ball-sized growths. Some couldn't even walk, so they languished and slowly died on the filthy concrete floor.  Sounds like pretty stock standard stuff that you see on factory farms. I wouldn't say that this is out of the ordinary at all, unfortunately. On April 28th, 2019, the Excelsior 4 and nearly 200 other activists descended onto Excelsior Hog Farm, conducting a meet the victims action. So a group of activists went inside the farm and they live streamed the cruelty and what they do is they do a sit-in protest and wait for police to arrive and part of the activists' requests are that the media come and report on the farm. So they generated a bunch of national media attention because of this. We start with the CTV News exclusive secret video shot inside an Abbotsford pink farm that supplies animals for the pork industry. And they arrested three of the activists, Amy, Nick, and Roy. Amy was the only activist taken into custody and charged. So what they tend to do is make an example out of the organizers. So they tried to deter activists by, okay, who's the organizer, you're coming with us. So a few months later, video footage from a hidden camera was released showing the owners and operators of Excelsior Hog Farm conducting criminal animal cruelty, including electrical prodding in the pig's face, repeatedly hitting and kicking the animals and cutting off the towels and testicles of screaming piglets with no pain relief. Wait a second, so the activists who exposed the criminal animal cruelty were arrested and the ones committing the animal cruelty were allowed to go home? What do you think about that? So basically they released this footage and an intensive investigation took place. So you'd believe that they're about to arrest the criminal animal abusers, right? No, as a result of the investigation, three of the other activists, Nick, Roy, and Jeff, were also charged. The Excelsior Four faced a combined 21 indictable offenses of break and enter and mischief with each break and enter holding the potential of up to 10 years in jail. After pre-trial hearings, Jeff's case was dropped. The remaining three of the Excelsior Four went on trial by jury in the B.C. Supreme Court and used that opportunity to further expose the rampant violence and suffering in animal agriculture and the complicity of our justice and enforcement systems. As a result, Roy was acquitted, Amy and Nick were convicted. They were convicted unlike Wayne and Paul who fought the case in a trial and were acquitted. These activists here weren't so lucky because it just kind of depends on the case, the judge and the jurors in the courtroom. Amy Serrano has made a statement on her Instagram here. Says, today Nick and I will be sentenced. I take responsibility for my actions while recognizing that without actions like these, the animals remain unseen with practically no protection. So they gave this footage to the BCSPA. Who are they? Okay, they're like the RSPCA, but for Canada, I guess. Listen to this. Jeff provided more footage of animal cruelty at Excelsior to the BCSPA, who turned him over to the police in violation of its policy. Am I reading that correctly? British Columbia Society for the Prevention of Cruelty to Animals turned over and activist to the police for exposing animal cruelty? Is that right? What scum if that is true? I can't believe that. Actually, nothing surprises me. The BCSPA, I don't know who they are. I mean, we could just only guess they're SPA certified. Yeah, so they do certify animal products. Okay, they've got an assurance. They've got a farm assurance scheme like they do here in the UK and in Australia. So they're in bed with the meat industry. So, okay, that's clear who's side they're on. It's only okay if the BCSPA go into a farm and find animal cruelty. You know what I mean? It's not okay for whistleblowers to find animal cruelty. The SPCA can call the farm and be like, hey, we're coming down. Make sure you clean everything up. Who knows, it probably happens, doesn't it? So they didn't have sufficient evidence to prove they placed hidden cameras in the farm. Placing hidden cameras in the farm where they're torturing animals and you're the criminal. So there's something called ag-gag laws, which basically prevents whistleblowers from sharing footage or gaining footage on the food industry, the animal abuse industry. Essentially, these ag-gag laws make it a criminal offence to expose criminal animal abusers. How's that? They don't want the public to see what these industries are doing to animals on their behalf. They want the public to see their propaganda campaigns, their humane slaughter campaigns, their humane farm campaigns, but these bills, like Bill 156, prevent the public from seeing the truth. That is criminal in and of itself. Now, who remembers animal rights activist Reagan Russell, who died trying to expose Bill 156? She was at fearman's pork slaughterhouse protesting with Toronto Pig Save and she was run over and killed by a semi-truck filled with pigs who was actually there to drop pigs off to be killed. And then we've got these four activists in Canada facing prison. The crown failed to disclose large volumes of crucial evidence. The Justice Verhoeven prevented us from showing animal cruelty footage or arguing that Excelsior committed criminal animal abuse. Now, the judge, definitely 100% a meat eater, prevented them from showing animal cruelty footage. Why would you want to do that? That's a very important point in the case. That was their motivation for being there, the criminal animal cruelty. So the judge withholds that footage from the jury to persuade the jury. It's like lying by omission. It's like omitting very key points. However, the crown's key evidence was my live stream inside Excelsior, which the jury saw. Okay, so they had to show the live stream because otherwise they couldn't convict Amy and Nick. So they showed the live stream, which contained the cruelty. Roy was acquitted by the jury that found Nick and I guilty of one count of break and enter and one count of mischief each. So basically they were convicted of break and enter. I don't know the details of the case, but essentially they walked through a door to film suffering pigs. So just imagine this. You hear children screaming from your neighbor's house, right? And you kick the door in to rescue these screaming children. They're being held hostage about to be killed or they're being tortured in there or something. Imagine a world where you got arrested for that. That's exactly what is happening here because the type of abuse and cruelty that is happening to those pigs is generally legal and it's illegal for these activists to go and expose it. There was a time where it was illegal to liberate slaves from the slave trade. In Germany, it was illegal to hide Jews from the Nazis. In current society, it is illegal to, in some places, expose criminal animal cruelty and to liberate pigs from suffering. The latest report I've seen is they've been given 30 days in prison. So they have to serve 30 days and have 12 months probation. They go into prison. This is actually quite a severe sentence. This is animal agriculture making an example of activists who expose their criminal animal cruelty. That's what this is because I know criminals who aren't activists who would have got off lighter, maybe got a suspended sentence, but no, they're going to prison. So they would have left. As far as I know, when you get sentenced, you go straight from the court, straight to prison. So when I got sentenced now, now, listen to this. I was a gang member. I was on drugs. I was carrying a loaded firearm.   People know about my past. I've been open about my past, okay? Turn my life around now. But I went to prison for carrying a loaded gun. Now, I had three assault charges before that. Three, where I got suspended sentences. So I didn't walk through those prison gates because of assault. And I got a suspended sentence for that, meaning like they convict you. They suspend your sentence. They let you go. If you mess up again, they'll put you in prison. They didn't get a suspended sentence. They have to serve a month in prison. So what they do is they take you straight from the courthouse.  So it's quite a scary experience. Now, these activists are going to prison, but the ones who are committing the criminal animal cruelty and violence, they're not going to prison. What they have is their justice. Putting people in prison who exposed their cruelty, so they win, basically. And these activists are going to prison for doing the right thing. If that angers you, if that makes you upset, then please go onto the Excelsior for website. You can sign, they've got a petition here. To the BCSPA, charge the animal abusers at Excelsior Hog Farm. So they're trying to get to 50,000 signatures. Imagine a world where you have to get a petition to pressure the Society for the Protection of Cruelty to Animals to charge animal abusers. You can also go and share the videos that they got from this Hog Farm. So they've got plenty of footage, share it fire and wide, expose this Hog Farm further. You can also donate to help them with their court fees and to help support Amy and Nick who are in prison and any of the campaigns they have coming up. I'm sure this won't be the end of their campaigning, but this is incredibly sad. It's incredibly scary to go to prison. Even me, when I was a gang member and I was protected as a gang member in prison, it's still, you know, it's not a fun experience and I don't wish it on any activist. It's just incredibly sad and frustrating because there was such a celebration when Wayne and Paul were acquitted and that sets a precedent for the right to rescue in the future. But here in Canada, they've gone completely the other way. What was the difference? You know, what was the difference here?  Look at what these scumbags did to those pigs. Look at what they're doing to them. These poor individuals suffering in hell, in darkness, in blood, in feces, suffering all around them, stuck, imprisoned in bars. These activists expose it, let the public know they're blowing the whistle on these animal abusers. Now, these activists have to spend 30 days in bars, no liberty, eating prison food with gang members and violent criminals.  What an absolute upside-down mess. If I got any of the facts in this case a little wrong, it's because I really just reacted to this and wanted to get this video out as soon as I could. Please help support. Go to this Excelsior 4 page. I'm sure there's plenty of contacts here. There's a contact here you can contact to help. If you're in Canada and you want to help, if you want to do any protests, please join their protest and kick up a big stink about this. This is absolute pure injustice. We need more activists. If you're a vegan watching this, please get active. If you're not vegan, look at what the industry are doing. They're leveraging the legal system to keep you in the dark. So you're giving your money to these animal abusers and they're putting anyone in prison who exposes them. Don't give them your money. Live vegan and vegans, please. Join the movement. We need more activists. Peace.


Main Street. It's not so very different from thousands like it all over the United States. Smaller perhaps than some you've known, yet large enough to service the needs of the local population. It has a modern drug store where you can find hundreds of items from aspirin and razor blades to coax and chocolate sundaes. As usual, it's the favorite hangout for the younger set who spend their leisure time indulging in laughter, calories, and that great American institution, the jukebox. There's a department store, spring fashions, a hardware store. They're offering a special on gardening tools today. A library, strangely enough, its doors are always locked. But there are many strange things about this town. Even a casual observer would note the utter cleanliness of the streets and houses. He would also receive the impression that everything in town was brand new, as though it had sprung up overnight. A closer examination of the inhabitants would reveal that all of them seem to be in their early 20s. Policemen, businessmen, mechanics. Even more unusual is the absence of older people and children. No matter where you look, there aren't any children. The population seems large enough and busy enough. He seems to center around the campus of a small college that is located in the center of town. As a matter of fact, the town is merely an extension of these classrooms. This is no ordinary school. The students here have been handpicked from thousands of applicants all over the country. Although they may come from different walks of life, certain characteristics are common to all. They are brilliant and determined. In the United States of America, you will find that as you drive from city to city, few Americans any longer use the centralized hotels which have been a part of their past. Instead, they use small places of residence on the outskirts of cities, on the main highways, which are called motels, motorcourts, and the like. Before going into this subject more in more detail, we will see a sharp training film in which you will see demonstrations of devices which have been used successfully in the past, as well as some of the latest models that are available. Take close attention. This is a typical American motel. They can be found all over the United States. Since most Americans use automobiles as their basic mode of transportation, motels have become very popular. Because of the luxurious facilities that most motels offer, even those who travel by rail or air seem to prefer them to the ordinary hotels that are found near transportation terminals. As you can see, the interior of this room offers many opportunities for the use of surreptitious listening devices. A few years ago, the only devices available to us were tiny microphones which could be installed behind the walls or under the floors. However, the installation of these microphones took time, and the accompanying wire made them susceptible to discovery. Today, we are no longer faced with that problem. To the unpracticed eye, there is nothing suspicious about this room. Nevertheless, these men are for all practical purposes speaking directly into a microphone. As part of the decor, most motel rooms display inexpensive lithographs enclosed in a simple frame. This one, however, is used to camouflage one of the simplest devices available. Enclosed within the frame, behind the print, is a complete battery operated microphone and transmitter. There are no wires or buttons visible. The batteries can operate continuously for more than 200 hours and are easily obtained. The advantages that this unit possesses are obvious. A new picture can simply be exchanged with one already in the room. Or if time permits, the existing picture can be utilized by inserting the transmitter kit into the frame. 3 fourths of an inch is the minimum thickness of frame that is required for installation. Reception is usually excellent, no matter where the voices are located in the room. A production figure for DC, I have the estimated daily production of three. What do you show unit cost at? The estimated unit cost, second one down, is $5,245. Not only can a conversation be plainly heard, but a permanent record made. One that can be used for future reference simply by using this small tape recorder in conjunction with the rest of the equipment. The listener doesn't have to be close to the transmitter. He can be a block away, sometimes up to a quarter of a mile away, if conditions are just right. A passerby would notice nothing out of the ordinary here. Just a motorist checking his route on a road map. Appearances can deceive. That hearing aid in his ear is actually wired to a tiny receiver which allows him to hear what is being said in the motel room. A permanent record of the conversation can be made under these circumstances just as easily as if he were hidden from view. The recorder can be hidden quite simply, as this man has done. There is enough wire on the spools to record continuously for four hours if necessary. Volume controls are simple to make. However, entry is not necessary in order to hear what is being said within a room. If the wall is so thick that the sound waves are blurred or otherwise indistinct, a long metal probe can be attached to the mic and pushed through the inner wall to the inside of the wall of the other room. The vibrations then pass from the wall through the probe to the microphone, clearly reproducing any sounds emanating from the next room. The contact microphone has other uses. Because of its small size, it is easily concealed and can be used even in public with success. It is a simple matter to lean against a telephone booth without attracting attention. The microphone is simply carried in a coat or trouser pocket and pressed against the glass. The conversation can be heard clearly by the listener. The microphone is used in conjunction with a receiver that may be located either on the body of the carrier or positioned some distance away. These are only a few of the devices that are available. Let's look at them again. The picture frame transmitter. It can transmit all the sounds in an average room as far as a quarter of a mile under optimum conditions. The contact microphone with different sized probes. It may be used with or without the probes and will faithfully reproduce all the sounds within an average room. Wrist watch mic. It will pick up a voice clearly up to a distance of 12 feet. The tie clasp microphone. Its range is also 12 feet. These small microphones will reproduce all sounds in an average sized room. This wire recorder can operate continuously for four hours. The small tape recorder, which can fit into a shirt pocket, can record continuously for one hour. These are some of the tools of your trade. Weapons of democracy. Learn them well. Opportunity to try each of the devices which you saw in the film. We'll take each of them in turn, learn what makes them work, and then back to the different installations. It's now 12 o'clock. Class is dismissed until 1. No, this is no ordinary college. The curriculum offers conventional courses in American history, geography and chemistry, plus some in photographic techniques, electronics, encrypting and decoding, and lab courses in the use of explosives. The students you see here had to pass some of the most difficult entrance exams in the world. They are a special group. They have to be. The rest of the town's population is just as special. The entire town is a duplicate of what you would find all over the United States. It has modern stores, familiar brand names. The menus offer nothing but standard American dishes. The theater shows the latest Hollywood productions, but it's completely isolated from other population centers and entirely surrounded by barbed wire and armed guards. It is located in the central Ukraine, one of the southern Soviet socialist republics. The entire town is a huge school designed to train Soviet youth how to live, think and react like their American counterparts. They drive American cars and obey American traffic regulations. They follow the latest news of America from sports to Hollywood scandals. Their English is not only flawless, but they specialize in dialects and accents. All of the inhabitants in town are instructors. They coach the students how to conduct themselves, even talk how to buy like an American. The only genuine feature about Winita is the fence that surrounds it. Winita, school for spot, small town, USSR.


This is your Current Ops Report. USS Porter is currently operating in the Black Sea in an effort to promote peace and stability in the region. The ship visited Odessa, Ukraine, where it hosted the commander of the Ukrainian Navy. The ship also conducted a passing exercise with Bulgaria, Romania, Turkey, and Ukraine, which included a live-fire exercise, visit board search and cease revolutions, and coordinated ship movements. I'm Petty Officer Corey Hensley and this has been your Current Ops Report.


In this time of advancing technology where most of us use our cellular telephones to text message, it was time for 911 services to advance to a level to allow our public to use that technology to reach emergency services and call for help. The slogan that we're promoting today that you can see on the poster behind me and you will hear several times today is call if you can text if you can't and it's a very clear message I'm hoping that all of the media that are here today will really impress upon our public is to use it just for that. A call is going to be more efficient it's going to be faster so call if you can. There will be opportunities where calling is not possible and that's where we're saying text if you can't call. Once again it can be overemphasized that the best way you need to call the best way you possibly can and that's a voice call but if you can't then by all means use texting. One of the reasons why that we were having this announcement today is to ensure that those populations within the three counties understand that capability exists but also understand that there are some limitations and that your expectations may not be met with text to 911. It's a best effort solution and as technology gets better then our solutions will be better but at this point call if you can text if you can. As this deployment moves forward we want to make sure that that everyone in the community all the agencies that are represented here know that we're going to continue to be here to help facilitate this process work with the the technology behind the solution to make sure that it's ready to go and to also answer any best practices and link up these communities with others that are already taking text to 911. I'll transfer you to CHP, we will ask you some more questions. So now what you see? XT test. Test. And we'll start from now. Okay, what kind of car are you in and you say? Basically, the speaker.


Let's pray. A kind and loving father we thank you for this day. Thank you for the gift of life. Bless us as we learn our children's amount for today. Help the children to understand what we are learning today and put it to practice. In Jesus name we pray. Amen. My name is Agneta. And I am blessing. Can you believe this is the last Sabbath of the year? And what a year it has been. We as the Glorious Coral we are thankful that God has given us this opportunity to close the year with you. Our theme for this month's children's amount has been The Lord's Supper, also known as the Holy Communion. We have learned so much haven't we Agneta? We have. Do you remember the shedan between Jesus and Satan in the Garden of Gethsemane? Oh yes. Christ sacrificed for our sins. It is the reason we celebrate the Lord's Supper. What about the sanctuary message? A timely message indeed. We learn that Christ is a sacrificial lamb. We no longer have to come to church with animal sacrifices like the Israelites. What about the foot washing ceremony? Jesus the brother. And last Sabbath we went deeper into learning about the symbols that are used for the Lord's Supper. Jesus the bread of life. The bread represented his body which was broken for our sins. And the wine represented his blood which was poured for our sins. Hmm after learning all that I think we have exhausted the message. Hmm now that I think of it I think we are in danger of forgetting one very important lesson. And what will that be? Let us revisit our Bible text for today so that we can understand our lesson better. It comes from the book of 1 Corinthians chapter 11 verses 7 and 28. And it says therefore whoever eats this bread or drinks this cup of the Lord in an unworthy manner will be guilty of the body and the blood of the Lord but let a man examine himself so let him eat of the bread and drink of the cup. So can you guess today's topic boys and girls? Let's say it together. Examine yourself. Actually the Lord's Supper is more of a question of who, what, why and how. I don't get it blessing please explain. Certainly. We have learnt who the Holy Communion points us to. Jesus Christ the Lamb of God who takes away the sins of the world. We have learnt what to do when we celebrate the Holy Communion. We show humility by washing each other's feet. We have also learnt why we celebrate the Holy Communion. We are remembering the great prize that was paid for our sins. Now we need to understand why. No, how? How should we partake of the Holy Communion? Is it really important? Of course it is. The observance of the Holy Communion is not just a ritual performed by our church every quarter of the year. It is sacred with sacred implications. So what are we talking about? How we partake of the Holy Communion is very important. Paul Apostol warned against the turning of this celebration to an occasion of eating and indulging. This is a time of suba self reflection and of our sins and Christ's sacrifice. So what you are saying is before we partake of the Holy Communion we should take a moment to confess to God any unrepentred sin and receive God's forgiveness. Exactly. Paul rebuked the church in Corinth for the application of the Lord's Supper. They did so in ananwadi mana. It says in the Bible in the Book of 1 Corinthians 11 verse 29 and 30 that those who eat and drink in ananwadi mana invite God's judgment upon themselves. For that reason many were weak and some were even sick. Some of them even died. Wow, that sounds serious. It's a big deal to God. It should also be a big deal to us. Christ's sacrifice should not be taken lightly. That reminds me many boys and girls got baptized in our church this year. They may not know about this. Boys and girls, the Holy Communion is a very wonderful celebration but it is not a time for fun and games. It is a time to remember the gift of grace that God has given us. Jesus Christ, our Lord and Savior, so examine yourself first. Confess your sins and repent. He had promised that if we confess our sins he is faithful and just to forgive us our sins. And you know what? I'm excited about the promise that Jesus made in Matthew chapter 26 verse 29 On his very first Holy Communion he said, But I say to you, I will not take of the Holy Communion, I will not drink of this fruit of the vine from now until the day that I take it with you anew in my father's kingdom. This can only mean one thing. Jesus is coming again. As glorious coral we are looking forward to the second coming of our Savior Jesus Christ. And we hope you are too. Let's pray. Dear Jesus, thank you for this day. Thank you for the gift of life. Thank you for good health. Thank you for letting us be here today. Help the boys and girls understand what you've told them. And help all of us to be here with you in heaven. In Jesus' name I pray. Amen.


Your coca-cola bottler presents Claudia Claudia based on the famous plays and novels by Rose Franken brought to you transcribed Monday through Friday by your friendly neighbor who bottles coca-cola Relax and while you're listening refresh yourself. Have a Coke and now Claudia What's keeping that wandering farmer you marry is the farmer yet. Thank you. You better be you like to bring home a new chicken If he doesn't hurry and probably just bring home the other half of the tickety almost got the last time he went driving Just because you're not along I suppose every state trooper between here in Boston will be waiting to pounce on poor David He knows we're waiting and he's probably in a great hurry. That's just when you get tickets hurry He's probably dawdling along pretending. He's driving a tractor. He's driving a tractor very hard, mama My experience with them is limited, but offhand. I think I prefer a Rolls Royce I think I prefer a Rolls Royce too, and it wouldn't even have to be a Rolls Royce. That's not very lucid, but it's perfectly clear David didn't say anything about tractors last night Does every farm come with a tractor or some come with mules? I believe David didn't say anything about mules either He hadn't seen the farm last night, but Roger told him all about it and Roger didn't say anything about mules I don't think they'd be what Roger would notice. Oh, David knew about was architecture Grand Old Doors and a floor that's a hundred years old Mama, what's so wonderful about an old floor? The beams are you probably hewn by hand? Oh, is that it? Why didn't somebody tell me that long ago? What's so wonderful about beams that are hewn by hand? Well, they'd have the eds marks on them That's terrible. Can't they do anything about it? Some people like them Some people like living on farm like it or not a lot of people do it well a lot of people live in cities And I'll see why we can't be two of them There are a great many advantages to living in the country Claudia. It's a much better place to bring up children You seem to be planning a lot of that I do see what's wrong with the city for bringing up children I was brought up in the city then there's a feeling of permanence a pride of knowing where you belong the dignity of owning the land you walk on and the Satisfaction of putting back into the earth more than you take out a fine one to talk if it's so grand Why didn't you try it yourself? I might have I mean you and father? We talked about it Do you really think you know everything anyone ever talked about I would have they talked about it to me You're very lucky to have a husband who talks to you about anything at all Some men would have just bought the farm and told their wives about it the day before they moved in David couldn't do that. He hasn't even seen the farm. He has now. I hope he had a good time I hope he won't be terribly disappointed if he's disappointed in this one. I'm sure there'll be another another I understand that most of the country around there is covered with farms I hope he won't be terribly upset if this one isn't what he thought it was going to be At least you'll know enough to be disappointed You don't know how most men get about houses. Nobody told me that either They think houses are something they were born knowing about like pocket knives or how to whistle I can whistle too I know I taught you thanks Most men buy a house and then after they've carefully installed their family in it They get the idea that they could have built it much better than it was built in the first place That must be a lot of fun. They proceed to treat the houses though.  Your husband isn't a man. He's an architect Besides mama, I don't think David really wants to move up to the country right now any more than I do He's just going to look he's taking an awfully long look. It's a good sign mama He really liked it. He'd have hurried back to tell us all about it But you better hurry back and take a good look at that chicken of yours. Oh I practically forgot I had a chicken find farmer's wife. You'd make All right, all right Well, if it isn't little boy blue How are the meadows in the court? Hello mother hubbard. How's your cupboard a cupboard? It's fair except for one right up old chicken a fine way to talk about your daughter. David. Is that you farmer david I can't hear you. I didn't say anything. I'll be there in a minute. I'm just basting the chicken What are you basting it to? I did hear that But I'll ignore it try and ignore this Oh, I can't But maybe if you gave me another kiss I could ignore that one No, good if you can't ignore the first time around it doesn't count. Isn't that so mother? I have turned my back and I'm not listening. I think mother's getting very old fashioned. Don't you?   Did you fall in some bottomless Connecticut marsh? Do you know what time it is? Is it six yet? It is seven yet. Is dinner ready? Are you been ready for hours? Oh, sure.  They're getting fuzzier in this hotel every day. Next thing they want me to wash my hands Mama, he looks disappointed to me. If he isn't now, he will be when he sees that chicken Should I ask him what the house was like? Whose husband is he? I don't think I ought to. Now that you've made me take off my coat, I don't feel like eating. See mama, he is disappointing Can't we just sit around and talk for a while? I wish Roger hadn't said how wonderful it was.  Did you have a nice day? It was a terrible day As you can see for yourself looking out the window. Well, maybe the house would look nicer on a nice day It would. I don't know. That's no doubt about it. Maybe you shouldn't make your mind up all at once Maybe you should take another look. I don't need another look. Oh, no. Is it fair to Roger to make up your mind just on one look? He made up his mind on one look. The sun was shining when he saw it. I don't need the sun to make up my mind Well, that settles that. Don't sit down here, David.  Oh, I feel like sitting down Oh, David, you mustn't be so disappointed. There are lots of other houses. Aren't there mama? So I've been told Mother, you've been misinformed. There isn't any other house. David, that is just silly. That doesn't sound like You mean you like it? Hmm. Something of the sort I think I'll sit down too. Tell us about it, David. It isn't just liking it, Claudia. It's something beyond that It's a feeling I got when I saw it Perhaps the same sort of feeling I I had that day. I took the sender out of your eye The feeling that this is the way things are going to be Without any warning at all suddenly I could I could see everything But David, I was me and this is only a house. Tell us more about the house, David. It's a pure salt box We don't have to go all the way to Connecticut to get a salt box. We can have it right here Well, they call them that because they resemble old-fashioned salt box. Oh, this house was built around 1760 Not all of it of course, but the main part the salt box I like the name salt box. It's cute. It isn't cute It's the valid appellation for that particular type of architecture. One side of the roof is longer than the other. Why? I don't know exactly why Sounds a little lopsided to me I don't know that I think it's such a good idea. It's been good enough for the last 190 years practically. Well, they are charming sometimes. This house is it's it's it's more than charming mother You can you can look at it and feel very close to the people who built it You can see how hard it was for them to build at all We'd say they they had nothing to build with and yet They built more than just a shelter It has it has a line Grace dignity Does it have much land land Only about 80 acres 80 acres That's an awful lot, isn't it or is it? How big is an acre? There are 640 acres to the mile the square mile it is 640 Is that a lot? That's a lot This is about one eighth of that. It's about as big as a Oh 10 big city blocks 10 blocks david that's tremendous. It's a farm claudia Not a window box and what land not not marshy bottom land, but but a beautiful beautiful slope with a Hill and a grove of walnuts on the summit and grazing land on the slopes and three farm fields and a meadow that Breaks off halfway down david you walked all over it. I didn't drive a tractor on a day like this If we lived on a farm you'd be getting your feet wet all the time. I certainly would there's a gravity spring and a brook Does the brook babble today? It was shouting. Would you farm it david? But not the cashew first We just put in a a few garden crops, but of course we had won a herd powers What do you think elephants could we have elephants? Well, not just as first my like elephants David you're really serious, aren't you about everything except the elephants Tell us more about the house. Well mother in the in the main part of the house The beams are all the original beam with ads marks. I should say so The four floor planks are are the old planks And you know that they have that that tonic color that they get as though they've been polished for years like an old shoe That doesn't sound so pretty some of the planks are even pegged. They're not nailed. It must be practically falling apart Does it have any stairs? I think the stairs are the best part of the house David, uh, are you going to have to do much to it? Well, I've been working it out in my mind on the drive home We won't have to do too much. How much is that?  Seriously david Isn't that going to be a big job? Not so big The important thing is the studying of the first floor and it's strong enough to take it. I love I know you look But isn't there always something that you find out about too late? Not if you're a good architect And mother we do have to have a nursery good girl Then of course some of the barns aren't in such fine condition barns certainly for the herd I want to lay in a concrete floor a concrete floor. Isn't that a terrible job? Mama david knows what he is doing Thank you, claudia You see I think we can lay it right in without disturbing the structures themselves. They're resting on a stone foundation But you can't be sure why not buy a barn that has a concrete floor mama david is an architect But he's beginning to sound just like a man. Then there are some small things We'd want to do right away after a while. We could uh, we could tackle some of the others What sort of small things? A few more windows and the annex and an extra bathroom But we could wait for that and we'd probably want to make a few changes in the heating system What if you have to do all that why buy this house? Maybe you can find a house you want to have to do anything to If you saw the house mother You'd have more answer than I can give you Besides it isn't as though david were just anybody who doesn't know about houses Have you changed your mind about farming mrs. Norton? You've changed your tune What tune you mean what mine mama means at a moment ago you didn't sound so enthusiastic And now you sound as though you love the idea. Do I? It's just what I'd hoped That you'd be as excited as I am And I think we can move in by spring move in that's what I said But david No I didn't mean to sound as though I loved it at all Leave here. I couldn't What make up your mind Oh Darling darling wait till you see the house that there's no hurt. I wasn't thinking about the house.  Why that familiar red cooler is even finding its way into your favorite food store Now when you're buying a carton or a case to take home You can stop in the midst of marketing to enjoy the pause that refreshes with ice cold coca-cola Every day monday through friday claudia comes to you transcribed with the best wishes of your friendly neighbor who bottles coca-cola So listen again tomorrow at the same time And now this is joe king saying olivar And remember whoever you are Whatever you do wherever you may be When you think of refreshment think of coca-cola More ice cold coca-cola makes any pause The pause that refreshes


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for introducing me. Yeah, so I know my name is quite hard to say, so my name is Paweł Borówka, which is Paul Blueberry, if you prefer. So you can call me Blueberry. Yeah, because in Poland we have this kind of funny names and surname somehow. So yes, okay. So the idea is to talk about the print varieties and the lecture by Angel Ortega. And I prepared some notes. I'm happy to answer any questions that you found during the talk and the lecture. And today and then tomorrow, I think, and on Friday I will have like one more hour. So please feel free. I think answering the questions is the most important part of the exercises somehow. And at least for me, it's usually that only like after like half an hour after the lecture, I start wondering about the details and I find some interesting questions. So that's why, you know, you have time to do it. So for sure, if you have questions, this is like a priority for me to try to answer the question, what Angela had in mind. Yeah, okay. So starting from the beginning, I just prepared the references because, you know, the first lecture was like very, very technical. So the references is the like the biggest or the best book in the theory of complex abelian varieties, namely complex abelian varieties by Bigeha Kalanger, BL. There is a lot of things inside. So yeah, so most of the first talk was can be find here and those are chapters like, I think chapters one to four, one, three and plus then the chapter, I think 11 about the Jacobians. Yeah, okay, so yeah. So if you find yourself, yeah, I want to say one thing that when I was a student, I was told that the theory of abelian varieties is this kind of a strange theory that you want to do some geometry and it's really useful to start reading this book, Bigeha Kalanger, starting from the chapter four and forget about, you know, don't read those technical details before and only if you really need them, go back and read them. Obviously it's not easy when you want to have a lecture. So that's why those first lectures on abelian varieties and primes are usually so technical, yes? And there are lots of questions and especially, you know, you shouldn't be, the people who don't know much about abelian varieties shouldn't be ashamed, it's usually like that, that you start and either you know everything and it's like boring technical stuff or you don't know anything and then it's like crazy magic in some sense. So that's the thing. We've prepared some exercises so you can find them on the Trieste webpage. I think I should, maybe I should be able to actually put them also in the chat or somebody else maybe can find them and put them. Download, there if I can, how does it work? File, so I will try to maybe send it. Yeah, so there are exercises. The bad thing about exercises is that they usually, we assume that you know what the prime variety is in most of them. So you need to wait for the second lecture of Angela to see the, to understand the question sometimes but I'm pretty much sure that there are many of you who already know what the prime variety is. So I really would like you to go with the exercises. In a second I will just try to say a few words about the exercises and what's going on there. Obviously, because this is not a lecture, I hope this is not a lecture but the exercises or the tutors or however you call it, the seminar, I would be really grateful if you start talking because communication is I think the crucial one in the studies. So if you have any questions please do not hesitate to stop me and to start talking. Yes, okay, so maybe I will say just for those who do not know much about a billion varieties like once more what's going on there. So, okay, so in terms of what's interesting about or what's interesting about a billion varieties and the line bundles on them. Yes, the ample line bundles. That it's really, that there's really the, you know, the CRM that says that if L is ample and L prime is a line bundle, bundle sorry for my writing actually. I usually try to write too fast. So C1, this C1 of L is C1 of L C1 of L prime, one and the second if and only if L prime is the translation by some point of L, okay. So it's really, it's really, you know, when Angela was talking about polarization, yes. In principle, she defined it as a line bundle. I prefer to call it polarizing line bundle because polarization is the chain class of a line bundle. But, you know, but the information is more or less the same, yes, you have a line bundle and then you can just translate it and this is the only thing you can do. So in the course, there was this question about this short exact sequence, yes. Big zero of X, X and then there's a very class of X, yes. And there was a question about if it splits somehow, yes, because there was, we wanted to talk about big zero to see that it is also an obedient variety, the dual one. So actually, yeah, it somehow splits. So it's not only that we have this, this statement that the line bundle is defined up to translation, the class. So if you make a little bit more technical fixing statements, then you can find the so-called characteristics and then you can somehow distinguish the line bundle with characteristic zero. So this X, this X, somehow if you have L and X, you can call this X, this X is a characteristic of a line bundle. A-ca-ra-ter-ish-t-x. And for principle polarization, this characteristic is uniquely defined, so everything is nice. If L is not principally polarized, that you have to be careful because this is the kernel of the polarization. So this kernel of the polarizing isogenic, if I want to say so, because you have these five things and you have to live with a thing that, when I say polarization, I may think of any of those five. So there is this kernel of polarization. So there are points X as that those guys are actually isomorphic to each other. So the characteristic is then defined up to this kernel. And obviously again, hopefully we'll not go into details, but if you have a line bundle, you can, ample line bundle, you are interested in the space of sections and they are defined by so-called theta functions. And if you have seen theta functions, it's good. If you don't see, it's also good because you don't want to see them. It's like a series of exponents. So there's lots of things which is in some sense hard because you are really in the analytic world. And for example, finding the zeros of a series of exponents is far from being easy. So that's why we really want to go into the algebraic point of view. And that's why the Jacobians are really important because for the Jacobians, we have this apple Jacobi map and we can think about this tori as an algebraic objects and as a divisors of degree zero. So, okay, so here, yeah, so here you can think about characteristic zero line bundle. Up.  Obviously. Sorry, but I'm not sure I understand really what do you mean by the C1 of L is equal to C1 of L time. So that they are linearly equivalent. No, no, no, no, no, those are all concentration classes. So this is in the Neurons' Severi class, yeah? So you can think about those guys as a this Hermitian form defined by the line bundle, okay? So this is in the Neurons' Severi class. Yes, so you're really mean equality. Yes, I really mean equality. Yes. Yes, so in some sense, okay, if you are here X, yes, if you are here and you go with the term classes and then you have equality here. And obviously, again, if you think about term classes, you can think of different things, but on a billion varieties, term classes can be seen as just the Hermitian form, positive definite integral on the lattice, yes. So you have equality here. And then you are wondering what's going on in the line bundles here, yes. And the thing is that it's really the same line model up to translation and it works for ample line bundles. So this is CRM, actually, I can check the statement. This is obviously in the Birkenhake-Lange book. Yes. Chapter for, I don't know, not from my head, but I know I can find it within like 15 seconds. Okay, so... Sorry, another question, sorry. Yes. And if you look at the first term class as the divisor, the divisor to which L is associated with, so what you get, you get exactly the same linear system on the a billion variety. So, okay, wait a second. So if you have a line bundle and you want to think about line bundle as the divisor, yes, then you can move divisor by a translation on a curve, on an abelian surface. And in general, those two things, so the divisor and it's okay. So wait a second. In general, they are not linear. They are not linear equivalents, yes. So in general, they are not linear equivalents. They are linearly equivalent if and only if the translation is in the kernel of this polarizing isogenic, yes. So they are only algebraically equivalent, yes. So exactly, this is, so in general, they are in this, it gives you the same class in the neuron savoury, yes. And sometimes, depending on the abelian variety, you have a linear equivalence. Okay, okay. So in terms of divisors, the second condition is the same for the associated divisors, right? Okay. Yes, thank you. Okay, so, okay, so maybe, okay, so this is what I want to say somehow about the line bandos on abelian varieties. And obviously, now we have the Jacobians, yes, the JC. So JC is, you can think about the... Wait, could you repeat please the last argument because I didn't get this part was X. So does this, the second part of the theorem pull for any X or just this chosen X? No, for every X in X. Okay, thanks. Because if you move by a point on an abelian variety, you get something which is algebraically equivalent. So this, yeah, for all Xs, if... No, okay, so there exists X, yes. In this statement, L prime is isomorphic to TX for some X, yes. Then they are in the sum X in X. Yes, so if you move by some X, then you get the same turn plus, the same hermitian form, yes. Okay, maybe now I will go into the exercises and I will try to persuade you that the exercises are interesting, fun, and however you want the exercises to be. And there are, I already, I want to say stress it. They are on different levels of... You know, there are some easy exercises and some harder ones. And actually there is a question which I don't know how to answer. So maybe you can actually help me to solve some problems. Okay, so exercises. So the first exercise is for people who don't know much about Apiglian varieties is just to show you that, if you have a complex torus with your lattice, yes, so C divided by lambda, then obviously you want this torus to be algebraic. So what are the condition on lambda and those conditions are called Riemann relations and you can look at it. Obviously for many people then you want to go into a construction similar to what you know from the elliptical, so-called Ziegels space. So you define the Ziegels space, so the matrices that we, which are special. So the symmetric matrices with imaginary part positive definite. So like a straight generalization of a Ziegels map a half plane and then you will know by construction that the resulting torus will be algebraic. So this is the first exercise. And then the second exercise is for you to see because okay, we haven't seen the definition of the prim map on the lecture by Angela but we already seen the definition of a prim map for double coverings on the Iran's lecture. So the idea is that, okay, so the idea is that you have the Jacobian here. We have a covering from C to C prime or from C tilde to C. So you have a Jacobian in the middle, then you have the smaller Jacobian here and then you have the kernel of this norm map, well known. And there are two, so the definition, the one of the definitions of a prim map is just the connected component of the kernel which will be one definition. But the other definition which you can think of if is by so called complimentary Abelian varieties. So if you have an Abelian variety, you can think, it's just a torus, so you have CG divided by some lambda. So if you find a sub variety, Abelian sub variety, so some smaller vector subspace, then there is a construction which shows you that there exists a complementary subspace. Yes, so this is quite a nice thing. Actually, this is technical to show, but there exists. And those two things will be called complimentary Abelian sub varieties. And actually those two notions of a prim, so one would be that it is just the zero of the connected component of the kernel. And the second one will be complimentary Abelian sub variety to the image of the smaller Jacobian in the bigger Jacobian. Those two things coincide. And this is exercise number two. The exercise number three is also very important. The universal property of a Jacobian, I will write it down in a second. And then exercise number four, this is again the classical statement of a prims. So if you have an etal-dabre covering, you don't, you know, when I said the connected component, this is really crucial because the kernel of the of a norm map has two irreducible components. And actually we can say what they are. And you have this description P zero and P one. So you have to show it. And actually you can generalize this statement if you have a finite covering between smooth curves, then the pullback is not injective. And especially this means also that the kernel is not connected if and only if F factorizes via a cyclic etal covering F tilde, I guess, of degree at least two. So this is also important because, you know, okay, so in general for the branched, for example, coverings, you have no problem. The Jacobian, the smaller Jacobian is embedded in the bigger Jacobian. And then you have the kernel, which is connected. And this is the prime variety as Iran said. But if you have, but if you have a covering which is not necessary branched or it is like a composition of two other coverings, then you have to be careful. If you have a cyclic etal coverings downstairs, then you don't have injectivity. Okay, six is one, the six exam, the six CRM is also very, very classical. So using Riemann singularity CRM, you need to show that the singular locus of the CETA divisors. So the CETA divisors on the principle polarized Avian varieties is defined, is well defined up to this translation. So the singular locus is also defined up to translation. And you can compute the dimension is G minus three for non-hyperliptic curve. And seven, I say seven is important because this is also what Iran was talking about also. So in general, the prime map is generically injective. But if we are in the locus of hyperliptic curves, we don't have injectivity at all. And this is somehow the exercise seven says this. So if you have a covering of a hyperliptic curves, that the prime map is a Jacobian of another hyperliptic curve. So you really, you know, you can think of it that the Picard group of the curve upstairs is built from the two Picard groups of smaller curves. And this is quite nice statement because it gives you the ability to work with those smaller pieces somehow. And yeah, and the question is how to find this curve and then if you find the curve, then you will see that the Jacobian of a hyperliptic curve is a prime variety for some covering and there will be lots of coverings that will give you the same hyperliptic curve. So you will have not injectivity. Exercise number eight, this is what I've said that I don't know the answer to. So the first part is in some sense easy. It's already, maybe you can find it also in the Iran stock. So you have, so why the prime varieties are also interesting because by the universal property of a Jacobian, the curve is usually embedded also in the prime. So you have this Abel prime map. So really thinking about the prime maps is also similar question. So the prime theory is also similar question to, especially in the very low dimension, namely for the surfaces is the same about asking what are the embeddings of curves into Abelian surfaces? So what kind of curves can you find in the Abelian surfaces? Maybe not necessary embeddings but by rational maps, yes? So if you have a genius free curve, you can embed it in Abelian surfaces, in Abelian surface, if and only if it is a covering of an elliptic curve. So this is a statement by, if the genius is two, it's already by barf. No, if the degree is two, it's already by barf. And okay, so there are some curves, genius free curves that can be embedded in the Abelian surfaces and others are not, they cannot be embedded. So we know that the genius free curve, a general genius free curve is a quartic plane, okay? So you have, you know, you start with your favorite quartic plane and the question is, can you embed it into an Abelian surface? So obviously because we are in the 2021, the question is, if you start with your favorite quartic plane curve, can you tell me if it is a covering of an elliptic curve 2021 to one? Because there is like three dimensional family of such curves in your, in the, in the modulite of quartic curves. And the question is, can you actually, you know, find the explicit examples, somehow explicit example of those. So, you know, this is something like finding not rational numbers, yes? Because obviously the covering for, you know, if this number was not 2021, but two, then it will be like easy because then you will have automorphism and so on. But for 2021, actually, if you don't like 2021, you can have anything which is bigger than what the, let's say 10, it will be already similar problem from my point of view. But it's quite an interesting question because on the level of Abelian surfaces, the question is easy, yes? Can you find a curve that can be embedded in the one 2021 polarized surface? So this is like an obvious question, but on the other hand, on the level of MG, I don't know the answer, how to say it. Yes, and then the ninth question is also like playing with those, with those prims. So in general, the classical prim is the two to one unbranched, so et al. And then two to one branched, but then you can work with any degree in some sense. And obviously, as Irene said, then you will almost immediately go out from the principle polarized world to the whole world of Abelian varieties. So the question is when the prim variety is still principally polarized. And there is a theorem in the Birkenhake Lange, which has a gap. So that's why it's interesting to do it because you will go to Birkenhake Lange, you will read the proof. And if you believe it, you are wrong. There is a gap in the proof. And there is one more case, which is not covered by the Birkenhake Lange. But actually it is filled by the paper of Lange or Tega, which you probably know. Yes, so that's the exercises. So I would like you, even like next to exercise sessions, I would like to find people who will be able to say a few words more or less, hopefully the precise words, how to solve these exercises.  Because Angela wants to go with a theory and play with prim maps and the geometry of the prim maps. But then those exercises are like down to earth, precise, explicit examples. So that hopefully some of you who don't know much about prim varieties will be able to do something or at least find some references for it. Okay, so that's more or less about the exercises. Can I ask a question? Obviously, yes. So in the exercise eight, you said that basically it's hard to find explicit examples, but we know that there is a three-dimensional family. Could you comment on why? So I imagine this does not depend on the 2021, this one for anything. No, no, no. Okay, so give me a second. Let's go back to my tablet. So the answer is very easy, yes? So that's maybe this. So the answer is very easy because you can see it from the other perspective. You start with your elliptic curve, okay? You take four points, you choose your degree, and then you know that there exists a covering C E with a branched D branched at four points. Obviously, this covering will be non-Galewa, yes? But it exists. So this curve by Riemann-Kurwitz will be a genius-free curve, which has such an embedding, yes? And now if you think about the norm map, you will see JC, this will go to E, okay? And then you will have the prim of C divided by E, yes, so yes. And then because this is branched at four points and 2021 is, no, it's not prime. I don't know. Who knows, 2021 prime? It's not prime, it's the first thing I checked. It's a product of two prime factors. Anyway, E is isomorphic to E, so... It's non-Galewa, you can assume it's irresistible, so it doesn't go via, because, okay, yeah, no, it doesn't go via any other, so you have like that. And then you have here P of C E dual. And now you can compute, so this is, the restricted polarization is 2021, because it's 2020, it's 2021, so you have here 1-1, so here you have 1-2021, yes, polarized surface, so because, yeah, because you have here a covering, so here C, which can be embedded in the Jacobian, will... Wait a second, am I sure that it will be an embedding or no, obviously it will be an embedding, sorry, sorry, sorry, this will be just a birational map, yes, because this will be my rational map to, so let's call it whatever, alpha, alpha of C, because it will live in the 1-2021 polarization, so the genus, the arithmetic genius will be 2022, but the geometric genius will be free, so this will have a lot to do, but the normalization will be C, so you have three dimensional family of those, yes, that's... Can you say why three dimensional, because you have four points of an elliptic curve, which has three moduli, and then you have one moduli for the elliptic curve? Wait a second, no, so it will be four dimensional, wait a second. No, no, it might be that I didn't do the deformation, theory calculation, it's just that the way you write it, it looks like four moduli, and then maybe there is a reason why one of them doesn't happen. No, no, no, no, no, it will be, yeah, it will be four, it will be four, I don't... Yeah, obviously it will be four, obviously it will be four, because yeah, we are in the genus pre-case, so any, okay, so there is one more, one more construction, we can go with like that, okay? So we start with any Arbillian variety in A3, yes, we can assume, so construct A that will be something like E cross X, so this is just one, this is like a surface, yes, so this is like E 2000, not cross, sorry, not cross, I want this to be this plus, so E that will be isogenous to the product, X 1221, so you have a moduli which is one plus three, I thought about the surfaces moduli, but obviously you have any E and any X, so this is four, and then by construction, you know that A is in general the Jacobian of something, yes, and because it contains E, with restricted polarization, you have this map, so again, yeah, obviously four is the correct number, I forgot about this one, so even better, you have six-dimensional family of quartic curves, and it's just to find something of co-dimension too, yes, and actually there are infinitely many of those, so it is dense somehow in the Zarisky topology, whatever, so you should be able very easily to find something, and it's really, unless this number 2021 is small, I don't know how to do, I don't know how to do, yeah, so that's the thing, okay, but we really went for the boundary of the questions of what we know, so let's go back maybe so that, yeah, are there any questions for the first lecture of Angela? I know, yeah, there was one question in the lecture, so I tried to answer with the splitting somehow, so yeah, and there was another question which I haven't heard because it was in the break, I was away, so maybe there are some other questions. Sorry, I have a question. Okay. How can we see that covering you wrote before it exists, this covering, it's easy, it's easy, but for... This is the Riemann's existence theorem in some sense, this is like... Oh, okay. This is the topological thing, yes, you take four points, you take the, you take the, you find the group in S 2021, which is transitive and you know, anthropologically you can construct it, so it's, if you want the reference, I think the Miranda maybe a good, no, sorry, Miranda goes with the P1s, but I think it's Yeah, Miranda, you use the P1, yeah, yeah. But still maybe, yeah, but I think somehow... Yeah, maybe I have to remove four points. Okay. Fundamental proof. Massimilial, okay, so the thing is that, okay, maybe, you know, if it does not exist, you can go to the second construction, which I prefer, yes? You start with any, with your jackal, you know, with any ABN freefolds, wait a second, what I wrote here, IA free, yeah, any ABN freefolds that will be a Jacobian and then, you know, for freefolds, there exists those guys that are isogenous to such things. And this is for dimensions. So those, because this is for, and this is for, it has to exist somehow, at least, you know, finitely many of those should exist. But I think it's just the Riemann's existence theorem with some, you know, possible changes, but I don't, yeah. Thank you. But it was a good question. So I think, yeah, I assume that Riemann's existence would be like enough. Okay, thank you. Okay, any more questions? Yes, I have a question. Okay, okay. Because Angela said that when we defined Jacobian, the map from H1 XZ to H1 XOX, which is by certain reality, same as H1 X Omega X duo, right? She said it's injective. But, well, this map isn't used by long exact sequence, right? From exponential sequence. And it's, I believe it's not injective in general. So it's true that you may embed, well, I mean, this homology is a lattice, but it's embedded as a full lattice and H1 X Omega X, which is C to CG, right? But this map, I believe it's not injective. For instance, I mean, you may have two classes which are homotopic, I mean, homotopic and... Wait a second.  Okay, you want... Wait a second, wait a second. I will just look at the, what do we have? Because what you're asking in some sense, if I understand correctly, is that the construction of a Jacobian, yes? That the Jacobian is the H1 of XZ. I think that's what Angela said is, is that the statement was correct and it is the Abel Jacobi CRM, but I'm not, wait a second. I just want to look at it. Where is the, okay. I will just go with them. Okay, so let me, again, share the, now this one. Okay, so unless I'm wrong, the Angela wanted to tell you this lemma 1.11, 11.1.1, okay? So that H1 is injective, or I'm wrong. And if you want, there is... Yeah, well, okay. Now it looks believable. Abel, can you just make it a little bigger so we can see? I would like to, oh, okay. Now I will be able. So yeah, there is... Thank you. Yeah, because I have the zoom, your pictures and it was, okay, yeah. So it seems that, yeah, it's, you know, this part is injective and it's, you know, it follows from the instances construction of the Jacobian. Yeah, so okay, so, you know, from my point of view, again, somehow the motivation of thinking about Jacobians was to compute those integrals, yes, so the analytic theory, but the Abel Jacobi theorem says that we are, we are in the algebraic setting, namely, that the Jacobian is already, it's also, you can think about it as a divisors, model of equivalence, yes? So this is very nice. So, but this actually, again, gives you the question, if you start with an Abelian variety, can you tell me if it is Jacobian or not? And this is the Schottky problem and it is still unsolved for the general dimension. So it's really hard question, again, as I told you, because this is like, you know, analytic versus algebraic world. Okay, so maybe I can say now a few, oh, if actually, if we are here, I can find the universal property of a Jacobian because we are already in the correct, in the correct chapter of Birka-Hakalanga, for example. Obviously, Birka-Hakalanga is not the, oh, no, this was a year, no, it's not. Yeah, so you can, you know, you can find a lot of things and obviously there are other books that are on Jacobians and curves. So, the universal property of a Jacobian, now you see the statement. So what's the statement? Is that if you have that the Jacobian is, it has universal property, namely if you have any other by rational, rational map from the curve to any other a billion variety, then there is an induced morphisms of a billion varieties up to translation because everything you have to in for, for a billion varieties, you either you think everything up to translations or you fix a point and you take, you put zero to zero. And again, by some lemma, this is more or less the same information. Either you have a homomorphism of a billion groups or you have any morphism and then by translations you get the homomorphism. Yeah, so the universal property of a Jacobian, why it's important? Because again, this is, you can think about the Jacobian or actually about the dual to the Jacobian as an Albanese. Yeah, so the solution to the Albanese problem as a habilionization of a curve somehow. So yeah, so that's the universal property of a Jacobian. Is there anyone who wants to say a few words how to see this, how to see the proof or maybe I will go. Okay, because today it's the first time and probably you haven't seen exercises. So okay, so today I will just talk but starting from tomorrow, I hope that I will find some people who wants to share some solutions to the exercises. Okay, so please, I really encourage you to say a few words about your favorite exercise from this nine and that would be nice. Okay, so in terms of a universal property, what's the proof? The proof is more or less, we take the Abel map from C to J. Okay, so the J is the peak zero and so we have to fix a point C and then we have a map from C to J which is just taking X to X minus C, yes? And this is the peak zero. So what's the, how to, so if you have a map phi, then the map phi twiddle is like, it can be defined in the only, like there is an only way to do it. Namely, you just take the whole points here, P one up to any point in the Jacobian can be written as the sum of P G points minus G, remember G is the genus and the dimension minus the G times C but C goes to zero. So you just, you don't see it. So this map is just phi of P one plus plus plus phi of P G minus G phi of C. Again, if you assume that C goes to zero here, you don't have any, you don't have this part. So you have something like that already. Yeah, so this is like the unique way to write it down. And the only question is if this is, if this is, if this works, if this is correct. And the idea is that firstly by rigidity theorem, if you have a map from C to X, which is rational, then it is everywhere defined. This is a special property of an Abelian varieties. So you have actually not by a rational map but by a morph, you have a morphism. And then again, phi twiddle in general would be a rational map and actually bi-rational because those guys are of the same dimension. But this is also a morphism, yeah? So C to the G and X are by, this is a morphism and here we have a bi-rational maps. And then, so therefore you can find this morphism T and the diagram commutes by the definition of this map. Now, phi will be a homomorphism because it takes zero to zero. And you only need to check that if you change C to some C prime, the map will still be the same, yes? And this is like a computation. So it's really the universal property is really not very hard to show. You need some properties of Abelian varieties and you need to know that C generates Jacobian and this is exactly because you have this Abel map. And I think you will see even in the Angela's lecture that the theta divisor can be seen as the image of WG minus one. This is also the Riemann singularity theorem. And so the universal property of Jacobian is says that what you think it's correct is actually correct. So that's the idea of a universal property of Jacobian. But it is very helpful because now if you have a covering of two maps, you have this norm map in some sense by the universal property of Jacobian. So that's the, this is how you can solve the exercise number what's the number, number three, yes? So the universal property of Jacobian. So obviously if you haven't seen the proof before there is some details to be checked from the Abelian varieties theory, but if you know the technical details then the idea of the statement is not very hard to see and to follow. Okay, so maybe I can just now stop sharing. Yeah, and okay, are there any more questions or maybe someone wants to say something about the exercises? Okay, so if not, maybe I will say like a few more generalities about Jacobians. And Abelian varieties in general for those who don't know much about the story so that you see a bit more. Okay, so as we said already, Abelian variety is C, G divided by some lambda, yes? So in general, in a very general, this torus is not projective. So we assume it is projective, yes, so projective. So in general, in general, the Neural Severi group of A is just Z. So you have up to a constant, you have just one polarization. So you are really rigid, yes? So there is this, so we have this line model L and what you can do, you can just take the cave tensor product of this line model L and obviously you can translate it by points in the Abelian variety. So yes, so this is one thing, which you should somehow know. So moreover, the endomorphism in general, the endomorphism of A is again, just only Z. So you can, what you can do, you can only if you have a point X, yes, you can multiply it's called, let's say N. You can take X in A and you can just multiply it to NX in A. And so in general, the endomorphism is also trivial somehow, so there is, and obviously there is this crucial automorphism namely minus one, yes? You take X to minus X, the involution, the minus one, okay? And in general, A is simple, i.e. does not contain Abelian sub varieties. Abelian sub varieties apart from the trivial one namely zero and the whole thing, yes? And this is obviously, this part obviously is over complex numbers, yes? Because if you think about it as a real torus, then you can always take it as S1 to the G and everything is not true. So we are really over complex numbers, yes, here, yes? So, and similar thing is for Jacobian. So yes, for general Jacobian, so, okay, so in general, so the dimension of the AG is G2 or G plus one choose two, so it grows quite a lot. For general Jacobian, the similar things happens. Yes, for general Jacobian, similar. So the general Jacobian, so the general Jacobian of a general curve is again simple, it doesn't have any endomorphisms and the neurons are very, it's also that. So if you have a covering, it's far from being general, yes? Because as we said, if you have a covering, then this curve is far from being general, especially your Jacobian contains a sub variety, the image of a smaller Jacobian, usually the Jacobian is also bigger. So we are really, you know, we're setting our story so that we can work, yeah? And that's the idea, yes? So then the Jacobian JC tilde, as we said, is the Jacobian of JC. Actually, okay, let's call this map F. Oh yeah, it's F already. As I said, some abelian sub variety and we will call it the prim, yes? So this is just the prim. And obviously we are interested in what can be the spring, yes? So as we said, for double coverings, the prim is of dimension G minus one, which we'll see the results of the Nagis for A5. For A5, yes. So then there is the prim map, which I guess will be defined again in Angela's paper. So if you have a covering, which will be in RG and then again, usually you define, you have some degrees and you have some branching because you can do it for any degree and any branching, and then you define this P of C tilde over C, which will live in some abelian varieties of some dimension and some, so here you have dimension and here you have polarization, yes? In general, for degree two, it will be like G minus one and the principal polarization, but in general, it can be anything more or less, yes? And when the prim map is injective is a good question or generically injective, it is very important to see that classically it's generically injective, but never injective. And as far as I know, actually, we don't know the locus of non-injectivity. So we know that the tetragonal construction gives you the locus where it is not injective, but we don't know if this is all or what are the other examples. So there are some still questions and some research going on when it's not injective, even for the classical case, but for non-classical cases, we know, for example, injectivity as already was mentioned in the Iran's talk. Okay, maybe that's all for today. I guess I should finish now. So again, I'm here for all workshop. If you have any questions, please do not hesitate to ask. And hopefully tomorrow we will go and try to work out some solutions to some of those exercises and maybe we will have some more exercises depending on what happens in the lecture. Okay, thank you very much. Last minute questions. If not, then maybe we have now our lunch break. Thank you very much. And remember that the gather is always open.


How do you decide when to go and is that is something that in communication with guys on the floor as coach Steph How do you decide when a little bit of everything a little bit of everything? But I think it's a feel for the game, you know You know guys got it going early Joe had it going so we want to kind of run You know, you know get him shots as many as he can and to you know, he can't shoot anymore Obviously Katie and Kyrie's gonna get it going at some point and then make sure in that second and make sure guys are getting shots And you know Reggie getting finished Jeff is getting in shots and open look. So just a little bit of everything You know, it's a game within the game