Welcom y top i TV er is ymwneud y gweithio'r ffynol gweithio'r seisio'n ddyn nhw'n ddyn nhw'n ddyn nhw. Mae'n ddyn nhw'n ddyn nhw. Ebbaton vs Bourmouth 480 o'r Sunday y Gwdysun Park. Mae'n ddysgu'n gwybod. Mae'r gwasanaeth yn gwybod yn ddyn nhw. Mae'n ddyn nhw'n ddyn nhw'n ddyn nhw'n ddyn nhw'n ddyn nhw'n ddyn nhw'n ddyn nhw. Oedd oedd hefyd? Ify roedd yr ytu. Roedd unrhyw warraedd nhw. Mae мерbeth i sunggu diwethaeth hwn digwydd yr unrhyw gwkeilio hefyd. Mae yna done ar l-lw Bernido. Rydych chi'n oswn gweithfa a legfosdu ako'r ysgan. Maen nhw qui trาย iawn amdalen consortio, parcell nad ddim eu wneud.谢 i'r senik fain ac er yw Body фильм. Robo'r hystarll o ran ddyn nhw. against the team, like, that has nothing to play for as well. We have been seriously lucky. Hang on, that fella on Twitter will have this down as a... it's a fix by the Premier League. I mean, last season, we were tuning down to a policy, we'd have lost that game, we were going to Arsenal to basically be delegated with me. We'd have lost that game, we would have done. Last year was essentially most win just like this. The palace game was basically, last year was wins. The Arsenal games are right off. This year, we've got the opportunity to win our home game and stay open. Ped, we've said, we have been saying that all week in our relegation battle therapy sessions, that we took this a few weeks ago that we had to win against in our hands. And if Nick Pope doesn't make that save on Monday night, then you're castle and injury time. No wonder you love the Pope. He's the Pope mate. He's the Pope mate. If that's the power of the Pope. Catholicism, the power of the Pope. No, it's not what we would have wanted in terms of our lives. But we went into the last few weeks saying, if we get our walls and we get our bone off and we're still in it, then that's good. Now, we've gone into the last game of the season and it's completely in our hands, completely. Yes, people can look at the game Newcastle less than go. Well, if only Newcastle won, then we don't really need a point. But that's a dangerous mentality as well, because it's great going, I don't need a point. And well, yeah, but whatever. Leeds won't beat spares and then suddenly leads are tuning up against spares and suddenly you've got to change every mental. Everton have got to win. We know that. The players know that. We've got to get ourselves to 36 points. And it's done. If we do our part, nothing else. There's no, it doesn't matter what the time and anywhere else. And that's it. And that's to me almost that's like a comforting thing, isn't it? It's like get rid of all the equations, all those other equations of you get pointing and they do this and they do that. No, we win our game 40 and nothing else matters.  It's been a season that could have been it could be almost predicted from the summit. You know, what we saw when we were out in America and what people predicted for the club and there was a lot of doom and gloom about Frank Lampard and being the first sack and everything. And to be fair, it's played out in many ways like that, but we wasn't the first sack. We probably should have been one of the first few sacks. And the solace I'd say is I look at this 17 at the moment and I do think they've got a little bit of something about them. That it is concerning about who's available and who's not available. But I do take solace that there's a few little characters in there. You're writing when the chips are down, we'll could rise to the occasion. Ty ach am i'n ped just, I asked just if there is about the season being obviously horrible and a disaster. It was almost knowing you get to the summit and the new season's approaching and when we were your age, it would have been can we win something, can we win the league, can we do, but obviously there's a chance of changing everything. But you have that optimism in the summer, don't you? Can we have a better season? What can we go on to do? What can we do now? Exactly in all that. But it was almost, we were in that summer period and because we'd lost Rytialis and he was the chief goal scorer for us, we knew we needed to add goals in it because obviously Dominic Alvedlun had had a great season in terms of being injured, you know, with his injuries and everything. We were looking for that goal replacement because the truth is even if Rytialis and Mother stayed, we wouldn't need another influx of goals anyway. You need to improve on what we were doing. Exactly the team didn't score enough goals and yet we didn't replace him and then Dom got injured. So we went from needing, definitely needing one if Rytialis and stayed so Dom getting injured and needing two and we only brought Neil Malpy and he didn't fill loads of people with lots of confidence. We were almost on the back foot right from the where goal last season with me. Or this season, I mean this season. Standing still in itself is bad because you can expect the teams behind you to want to move forward. So then to arguably move backwards and you know, we did bring in McNeill and you know, if he hadn't over performed in terms of scoring goals, which you know, he's never got this type of goal tally before. And I don't think he was brought in expecting him to get this sort of goal tally for a decent player to add because he's got his strengths and his abilities. But he's never really meant to be our goal scorer and saviour and now he's our top scorer. And if he hadn't performed for us, especially the last few weeks, the way he has, I dread to think he'd be in and that's not because McNeill blew everyone away, but he stood out in a very poor squad, hasn't he? So to, for him and Malpey, two players who've never been create a goal scorer, you know, I know Malpey got a good record in the championship a number of years ago, but ever since then he's been the one who's at Brighton. Everyone says, oh, Brighton just had the proper striker. They'd be on to good things and he was the striker that they had on everyone. The second Brighton, the real striker. So yeah, it was a mad one and I feel like a lot of us just sort of skipped that some of the optimism that we normally get. Obviously you'd still get a few. Remember speaking to people who thought we'd get top half if everyone stayed fit this season? I think, you know what, don't you? Sorry, Peck, just where you're going. But I do feel like if Don would have been, you know, and people can scream at whatever you're watching her on here and go, well, it was never going to happen. But you had to hope that he was going to stay fit and if he would have stayed fit, I think everyone would have been halfway up. I really do. I think we proved at times in the past few weeks that when he's there, the team isn't that bad. But our attacking offerings, and you're absolutely right that we were, I remember Peck saying a few weeks ago, we need someone to step up and take the mountain. And it has been, Dwight, with Neil, fair play to him. Started the first half of the season, he was poor, he was. Sean Dight has even said the same thing that a fella he'd given his Davey. But since Dight's coming, he's got better and better and better. And he's weighed in with these goals and so much so that we missed and higher up the pit against Wolffs didn't we because he was having to play so far back. So, hopefully, on Sunday, and we'll get on to that. I mean, it'll be a bit higher up the pitch, but we have needed someone to take that mantle. It has just been one thing after the other. And paired it again, we can keep completing. Can I just go back to the dominant camera and point, because I think it's really interesting. We're going to look at the whole season in a way. It's out. Mopai, when he signed, had a record of not too similar, dissimilar to the Charleston when he played for Brighton. 8, 9, 10 goals a season. The problem is, all season, we have stuck to this idea that Dominic Avalon was on the way back, literally all the time. So, we have not evolved the team to play in a style that helps Mopai playing. So, the style has always been to play for Dominic Avalon when he wasn't playing. So, you go along stretches, six, seven games, eight games, whatever it was, without Dominic Avalon. Even more, where you weren't playing to Mopai's strength. And I feel sorry for Mopai, in a way, that once again we brought a player in who isn't anything like Don, but never played to his strengths once, even though we knew that we'd have large stretches of time. What are his strengths, though? His strengths is coming deep, taking the ball, and then, either training or starting the attack deeper and then other people run and pass them. Now, we don't have a team set up for that. So, that's the biggest problem.  And we haven't. So, we've been stuck in this Dominic Avalon limba wall season of waiting for him to come back. And the simple truth is, we'd be better without him because we'd be better moving on from. He's a fantastic player and I don't hold him any personal responsibility for his injuries because I think he wants to be as fit as he possibly can. But this is a football club that couldn't look past them and it's caught us out massively. And we shared in the first week of the season when there's still time, you need to go and get a Dominic Avalon limba wall replacement. And that did not mean going by a £60m strike. By the way, that was saying, we need a guy up front that we can rely on for the whole of the season. And when Dom comes back fine, we can put him in. But the pressure's being on Dom all season to come back, come back, come back because of that. And that's why we find ourselves where we are. We could have bought a player with maybe a similar skill set but nowhere as good. Or someone who didn't have any skill set near him, but you've honed your team to play for that one player. And that is awful recruitment and management by everyone involved because they've gone, ah, this fella, this fella, he'll be back soon, he'll be back soon. And we've completely destroyed our season and actually our best player using that kind of absolutely foolish mindset. I was going to say, is Kevin Fellwell and the recruitment team to blame me or as well as Frank Lampard because he was in the building when we had a situation with Garrassie who was now playing for Stuttgart who basically his team in France allowed him not to train for four days because Evan had done the deal. And for whatever reason, it got pulled at the last minute. Now, he's not amazing, don't get me wrong, but he's six-foot-three and strong and powerful, 26 years of age, can move, he's moving around the pitch, he's good in the air. Very similar to how we would play to Dominic Calvillan. Dominic Calvillan's a better player than him, let's have it right. But what that would have enabled us to do and he was 11 million, I think, Evan had done the deal at 11.5. It's about the same as how we would play to him. Only just suited Evan more. And this is a player as well, by the way, that if Dominic Calvillan would have then come back fully fit, it wouldn't have been a big uproar if he would have then moved to being on the bench as well. And also, you could have managed him through that then and gone right. We're giving Dom 70 minutes, then he's off. Oh, Garrassie's playing the face. We've got a midweek game. Whatever it is, I've developed a both because that would have been like for like. And yeah, Peds right in a way. We've people are having a go with Malpie. Malpie, for me, is an issue for the simple reason that he's not, he doesn't stay up front anyway. So he's like, he's almost like a number 10. And if Frank Lampard would have said, we've bought Neil Malpie into play with Dominic Calvillan, he's going to drop off him. And they're going, oh, all right, I kind of understand what they're thinking of. But he said to us when we interviewed him. He said to Sky, he said to Everton TV. No, that's not the plan. So you can't go with an outball to Don in the air one week. And then play the same outball to Aladdin's massage and expect me to be winning it. The centrefacts are six foot three and four. To me, that is bizarre planning. And that's one thing that has got to change. It hasn't football. There's nothing wrong with having a plan being a different style of play, but we never got that. We got the player for it. You know, you can say maybe it wouldn't even be great if we had that style. Maybe you'd get seven goals. That'd be six more than he has. That could mean a hell of a lot more points for Everton. If we'd done that.  Then we have our style of play and we can replicate even if our best striker isn't available. You mentioned Kevin Fellwell. And Fellwell's one I'm very unsure on. Because obviously Wolves, when he was the director of football, they made some very good signings, but you've got ones about his involvement and how many of them were agents. Fellwell have signed them if he had the chance. Would he have liked to sign them anyway? But would he have not been able to if Mendes wasn't bringing them in? And then the other thing for me is as well, we've had two directors of football before them. And now they both came with their own faults. We can acknowledge that. But we also know that they also had their abilities hampered by our board. They weren't always allowed to do their job the way they wanted. And you've got a thing. Can three different directors of football really fail? Can they all be that bad? And maybe they could. It's something time will probably tell us. You know if things change on the board and then maybe we'll be able to clearly see Fellwell's influence more. It is. Don't you think though it's math and you're not disagreeing? By the way, sometimes when I've said this about players, when managers have failed so many times, it's like, is it always the manager then? Is it really always the manager? Some of these players have had seven, eight managers. And we've had all different kinds of managers. So I think sometimes you don't blame love saying it doesn't have business as fail managers. But is there a situation where you're looking and going about the likes of Kevin Fellwell and saying, well, what's he saying? Does this play? Well, we know Bamba Deng was another one. He's over six foot. They haven't done the deal with him. And he got pulled. He agreed he was getting on a plane. He got pulled the next morning. And as it happened, he ended up failing the medical it leads anyway.  But is it like one of those things where Kevin Fellwell, Dan Pedy, whoever it may be says, we've got this player here. Ibrahim Singari. Ibrahim Singari. And I know that his brand was there. Brands say, we want to buy him. And someone on the board goes, nah, he's not played in the Premier League. Do we believe that? Where's that coming from? Is it the manager, do you think, who's saying that? Or is it a board member? Is it far out of Marseille? Because I can't believe. I sit here and mean, Andy, the pub have done it loads of times and do it now. Or I'll just have to be on the chat with Perth over there and go, seeing this kid here and, oh, see, it looks really good. These are only available for this, this, this. And I can come up with players and I've done loads of players on here who haven't could have bought for a fraction of the price and they've gone on and done really well. Singari being one of the 8 million quid, talking 40 million for them. I can't believe evidence recruitment team with all the resources they've got and all the data and all the professionals that are there and talking about the same people I'm talking about. I'm not having it. They quite clearly are and quite clearly know everyone. So what is stopping that from them identifying and saying three strikers who are over six foot, right? What's stopping today? I'm here when they join evidence. What's broken in the middle? Well, I think you could go back in time over the period of a few years and pick out five great players that you've mentioned or someone else has mentioned that evidence should sign and that they have been important and that would have been good signings. And I think you could pick out five players that have probably all been vetoed by five different people over time. For one player, you might get the manager saying, I don't want him, he's not my style. I want to play football this way. Over time, you could get the owner saying, no, we want someone with a slightly bigger name. Maybe Bill Kenwright, for example, says, oh no, what about Premier League experience? Maybe one of them is a younger lad, maybe goes into under 21s for a year first and David Unsworth says, no, not my style, I don't want him. I think because we don't have a clear defined structure there's probably a lot of people saying different things and over time we've seen a few get turned away by different people. There's too many cocks, isn't it? There's just too many cocks at the football club. There's too many people who don't know what they're doing and they clearly don't. You've got the old structure with Bill Kenwright, you've got the Mishiri coming in with his ideas, you've got his people in his ear telling him then you've got the directors of football then you've got what the manager wants. You've got no defined system. We have to change that. The new clubs now, Brighton, he will be bored of some of Brighton's book and I understand the thing of like, well, it's Brighton this year and it was Leicester last year or whatever. It doesn't really matter, it's like someone like Brighton or Brentford let's just use those two an example. They've got a defined way of doing things and you don't ever hear. There's no chaos, there's no drama. If they fail, they fail because football fails sometimes. You don't win a game of football because it is so you don't win a game of football. Well, they simply deployed and went after it just doesn't work out the way it is. But there's a defined way of doing things and you look at how things get better at those clubs every year because the system's working better and because they've got more experience in the Premier League and they're attracting the place that they actually want.  Brafford Bonita's coming and brought his own recruitment team in when we had the director of football who just signed a new contract and slapped him in the face for him. Would you like them or not?  Man's come out and said this. Man's come out and basically said I was born on Garethman leaf, the day Rafa Marisa's come in. And it was like I didn't have the agreement. The problem with that started when he sacked Michael Silver and it was him again. It was the owner sacking a guy without the permission of his director of football. It's like this, we are at the bottom now. We are at the very bottom and all we can hope for and this is a match review, by the way. All we can hope for when we go actually in this matches is that it evident survive that this is the turning point. The turning point should have been 12 months ago. It really should have been. They should have been resignations the day after the Arsenal game. But we were all exhausted and I think a lot of people got themselves to believe that that was the start or something because of the unity within the club because people liked Frank Lampard. But what it actually was... But it had been fracked. But what it was, it was just Emperor's new clothes though. It wasn't real just because you like a manager and he gets the pie out. Doesn't mean anything's changed and the club played on that. And that's another reason the all needs to go is because they abused the fan base by, they used that against us and they're still using it against us. There's images of the fans outside the ground and the pitch invasions while telling you not to do any of these things they use us all the time and obviously that all comes ahead in January. They try to use us for one other thing. And that all, this is now, we are at the bottom if we stay in the Premier League because this is our lowest I think we can look up. But obviously if things go badly then we've got another set of depths to drop into and that's why this game is so important but people have to understand that. This didn't happen by accident. Last year you can go well that happened by accident Carla went and this happened and that happened and we got all these injuries and how did we know that Dominic Arvenloon with Charleston this player, that player fall out with Dean all this was going to happen how did we know? You can accept that for one season but for two seasons this is when people talk about managed decline this is managed decline this is, you know you sell your best players you play some of the best players what do you think is going to happen? This is what happens. Moving on to Sunday. It is a massive preview. Obviously it's a huge game one big issue is obviously Dominic Arvenloon who spoke about him before but he limped off before half time he's out of the game and all that so what would you be doing in the forward positions? Is it Malpie? Is it Lela Sims who's obviously hasn't started the game of Cullwsian Park all season or is it Damari Grey who managed to turn to a few months ago? Probably I'm looking towards Damari Grey I would like to see Sims in there I don't think I should play him but also we've got the side of well you know maybe you want someone who's a little bit more experienced than the team for this game so I do take that so I think Damari Grey for me probably is the option if not Sims is the great class in terms of emergency to bring on as well I'd definitely bring them on before Malpie but for me I'm looking at probably Damari Grey up front and I'm asking to Corey to keep in that advancement field role be less sloppy on the ball than he was last week but you know make his runs into the box almost players a second striker at times because that's what he's good at Well that's I mean Pedd he has popped up with some important goals with like the Corey and he is hot and cold and you see that with him but he is probably the only one of our midfield players who's likely to break into the box and get on the other side and I think what he gives you is he gives you that physical out ball of times if you know that he's going to have to he's going to have to be more advanced I think what's important is that no, no week to time the camera room wouldn't be ready for this game so therefore you can look at different ways to me that would be Damari Grey I'd put him up front I just don't think Alice Sims has got what is needed for the game I mean if he wants to prove me wrong on Sunday he can happily prove me wrong but I think Damari Grey having an out ball that could stretch them stretch them so that the Corey can fill that space and then maybe be the out ball when we need it we've had Damari Grey's played up front before and has worked hard he has actually been selfless which is people have remarked maybe not one of his great straights so yeah I think I don't I think what Damari Cavalun has brought us was the ability to bring other plays into the games without scoring goals himself because he hasn't really scored goals and that's what you need you need to centre forward who's going to occupy and then that allows those spaces to open up for the Corey or a Mach Nill or a Wobby or whatever and that's what's going to be needed and we just have to put ourselves in a place on Sunday where we're dominating swarming where we're not giving them any breath so that that actually means that you're not playing one up front that means you're dominating and that means that you've got the Corey pushed on we've done this haven't we Brentford at home was one I can think of to drop on me ever and we didn't have Damari Cavalun and that first half the game should have been put to bed we absolutely destroyed Brentford the second half was a different story and we were hanging on a bit but the first half we should have put the game to bed they let it out first half should have put the game to bed and didn't it too full them once we equalised we had the chance to put the game to bed and we didn't and then it's flipped and that's what we've got to get right Sunday what else though tiach obviously no right back obviously Patterson's or Coleman's mission Ben Goffrey's touch and go but whether or not you think it is the decision because it looks like it looks like Michael Ancoe's got a chance a good chance of being back which means that we've got to left back again but on the right what are you thinking here is it Mason Holgate is it Conor Cody to play a little bit out of position or is it a case of because Damari Grey's going to play up and if Michael Ancoe's back Dwight McNeill can go back to where he's been which is on the left that means that it's either a woulby or James Garner in the middle if Oana comes back into the side I mean started last week he's got options hasn't he because obviously James Garner when I played right wing back and right back for a bit last week does he go James Garner right back in a woulby right in the field or what would you be doing there I wouldn't personally be against seeing Garner there again I think he did all right against Wolves the only option you're looking at as hard as that effect in the middle of the pitch especially if to Corey he's pushing up obviously you've got Gway in there and maybe Oana comes back in as well and the other option is to play a woulby there and ask him to play as a right wing back but then maybe you're operating with less width because you don't have a woulby on the wing and if we're talking about stretching if I'm making the entire pitch and suffocating them off the ball having players in every position to press their full back stop them getting the ball out I do think you probably do need to be up there in a wide position and you can do that from wing back or on the wing but I think the safer bet is probably a woulby at right wing back and I probably would go to James Garner add a right wing back again and go a woulby right wing so right back and right wing in the floor what would you... Tom Davis Tom Davis right back come off the alley if he's not fit which he might not be fit according to his own dice break my card or would you go to three at the back play me to an all gate play me to an all gate play me to an all gate I think you know listen they they've got pace why I haven't eaten typically I'm just doing your standards last please don't play what's your goal against them always goal against them I'll say I'll just play me to an all gate there is a temptation to go to three because Cody's available but even on Saturday where he could have gone to a three he didn't go to three with Keane coming on the pitch so I'll put me to an all gate right back make a link hopefully drops in at left back and we can keep things how we want them then we've got so what are you doing in midfield then because obviously I'm not and I was in last week when James Garner was playing different because the right wing was fed up actually it is a difficult one I don't think you need to overthink that I don't think we need to overthink it I don't think it's either Garner or O'nana I don't think we can sit here I'm really I can see people looking at James Garner but I'm being reliable informed this week there's stats on all different from Tom Davis but when he plays there's a stature to him I understand that he uses the ball well but that's the way though busy busy can be something that isn't real at times in football I think O'nana can pick up the intangibles no but sometimes the intangibles don't actually mean anything that's the point O'nana can look sometimes like he's absolutely terrible and then he's you look at his stats and his stats look like because he's winning tackles, he's keeping the ball but he's playing it at his own pace so O'nana gives you that physical element and that might be needed because there's no drama of government I think it's a flip of a coin it's whatever this week on 16 go we understand what the tactics are going to be and that will come down to what the manager decides tactically this is the evidence he started at Mollingew last week and obviously that's going to change this week because we don't have Dominic Calverlun and we don't have Nathan Patterson it will be on in a sec I think Patterson mean it's our cost you must nail down O'nana the core of Calverlun so we know two of them are going to be out which is Patterson and Calverlun and that was a a 4-5-1 formation but that quickly changed to and ended up being I don't know what kind of formation I don't think we had the formation chaos we just did it was chaos it was like a what was it, Hail Mary was it in the American football John Madden's when he just caused mayhem it was like that but if we look at Bournemouth this is the side from last week's game against Manchester United to see how they lined up Neto and Gold, Smith, Zabirani, Sinesi, Kelly David Brooks great to see him back Lewis Cook Christie, Lehme Anthony and then obviously Solanki as a long striker and then they have got plays on the bench you can cause your problems as well they had gone back to they played almost a 4-5 on the times they played 3 this year and I don't know what we're going to get from them I'm not going to say these are on the beach because they probably won't be but what I would say is wherever maybe can exploit it with the crowd and with the players mindset is that no matter what Bournemouth won't be at 100% because you've seen it the last you've lost the last three league games since they were safe it is in the cage of it it's not like a thing you're going oh lads this doesn't matter today I don't think there's that question evidence sometimes but no team does that but what you sometimes struggle to get when you're a team with nothing to play and we've seen this a lot in those heady days when last few games didn't mean anything is your team can just drop below the required levels and if you're playing against a team who has got it all to play for and is fighting and scrapping for everything there the team that normally comes out on top and I think that's what and I've got a hone in on on Sunday and I think the crowd won't make it like that but that's what we've got to look at because I don't think it'll be flip flop time for Bournemouth but we have got to squeeze that extra 2% or 3% out of ourselves that could get us over the line because these aren't man's city are they so? No and correct me if I'm wrong here but I think they've actually been losing games since before they were mathematically safe I think you know they were safe but I think neither the point against Chelsea to confirm it and they lost that day so they've lost the last 3 obviously not flip flop time the professionals as you've said on the live before there might be a couple of players featuring here who may be looking at a move and the summer maybe rotates the squad slightly or the players looking to impress the next season David Brook started his first Premier League game in a number of years last week obviously he's going to want to build fitness he's going to get back on track he's going to be in the team every week next year and he's a very good player very talented player and I don't think they're just going to roll over for us at all because why would they but what we can do is we can reach greater levels of desire than they can because we have something to play for they have 3 points to play for we have one of the biggest games in our history to play for which is where we can get that advantage and that can also create nervous energy as well and that has to be managed but if we'd handle it right and we use it correctly and we play the style of football we're all saying that evidence should place really suffocate them that'll go in perfectly with what we're predicting the atmosphere will be like a hostile atmosphere for Bournemouth when in a situation where they don't really have much to gain I mean and we can look at this I know it was at the end of the Covid season but and they ended up going down that day but they came to go doesn't know in a deep one they had an outside chance well they had a chance of staying up until Jack Greelish I think it was school for Villar and kept them up but they beat us 3 when we were at the end of the season Carlos first season we couldn't do nothing but it was a game we thought we'd win at home against the team going down we had a decent side and all that we took the lead on how we equalised mwych geni equalised but we ended up losing the game 3-1 and sadly for them they were still relegated but their desire that day meant it was a bit more than us and they beat a better team as we were halfway up so we were a better shot that day with that desire and we have to harness that the other way this week don't we we have to be the ones with the crowd begging for blood making it uncomfortable for them we've seen the Crystal Palace players saying it was horrific playing the goodness in last season it was horrible in last ten minutes they just wanted to get off the pitch because they were what's going to happen there and they could feel it in the crowd and that energy obviously was the other way with the other players who grew on to the win we have to be on that right from the start on Sunday and they haven't had the gut to feed into that having the players we've got to make sure we don't being ourselves out of it the emotions don't overpower us as fans as well it's going to be it's not going to be a nice experience it's not going to be a nice experience we've got to make sure with the emotions just don't get on top of the players and just try and play the game just try and play the game like it's you know like the managers told them to stick to a game plan don't get it don't do anything stupid don't react to anything stupid for the fans whether there's noise or whatever it's about the game plan it's about being professional that's all that comes down to yeah the difficulty is the players having a mission it's not a case of going oh my god these are amazing, it's a look at it and going we've struggled our own to score goals how can we change that but you've had a whole week to work on that haven't you they'll have known, Dominic Arwyn got injured Saturday they'll have known all week so we have to look at other ways how can we get it behind Bournemouth and that's the key the thing is as well what you mentioned there about temperament being important and managing the expectations and the emotions that is why it appears when I went through right back I don't know if it's temperament maybe but he's still an experienced player he's 27 he's an experienced player he's got to have his head on it but you might be right it's just that they're not amazing options we're not blessed with opportunities he's the only one I look at who has played right back started his career at right back knows how to play right back and the others are just feeling it and that's the problem or you change the shape as I said I'm not going to sit here and go you've got it completely wrong you've got to have an element of trusting your manager especially when you haven't got players the two big things for me are one it doesn't actually matter what's gone on this season for this game whether you think Evan can score goals or he can't or whatever we've got two goals of forest we've got two goals of Stamperbridge we can score goals without him we've had to most of the season he's played like quarter of the game but he's got it and we will get chances and we're going to have to take them we went down to Brighton the other week but he didn't score a goal and we took five chances we almost had another one so we know we can score goals so we've got to believe it doesn't matter and once that whistle goes whatever's happened in the season doesn't matter it doesn't really matter it's about that 90 odd it'll be 100 minutes because referees love the island you're eating so it'll be that and for that period we've all as a fan base have got to be on it the players have got to be on it and get through that and if we can do that I hope when we're successful at the weekend there isn't a pity in basin there is everyone's stage where they are and then we do whatever we celebrate every leafness and then people can either chance about the board or they can go and have a pint with the mate or they can whatever we just keep it managed but within the ground because I honestly, the Premier League are looking for anything and the papers, the press are looking for anything so let's hopefully for me and it's only my opinion not telling people what to do but I hope we do keep a lid on it as in we stay where we are be noisy and vociferous and everything but to stay where we are and I think if we play it that way certainly the first part of that definitely we treat it like a cofan to one off not matters, support everyone of own honour it's two passes out of play so where if we knock it back to pick for the chance it out for the throwing still stick behind them because it's our football club some of these players you might want to breath might walk out the door in three weeks and it might just be a thing that was in their life it is, us it's our lives isn't it and we have to remember that there's a chance for one of these players to be a hero as well for one of them to be someone people tell the kids about in a few years time when our nan had us up when he stood up and he was countered and he led by example and he played well and whether that's o nan er o someone like him, yeah there's a lot of noise around him he might move on in the summer well if that's going to be the case then be the kid that steps up like you said try to get yourself a couple of goals and go like the charlison did last season score the goal that keep you up and then go there you go that was for you that's me done now but I've gave you everything I had it's a way to bow for both of us now I'm going to I mean it's another one isn't it it'll be his last game for everyone Sunday and he asked give everything go out with a big bang score a couple of goals and keep a clean sheet go out that way you know what I mean be a hero and hopefully this is the hopefully this is the start of better time for everyone football but it's a huge game, it's a tough game it's going to be horrible hopefully from our perspective Spears and the charlison plays another big role in us staying up and going and does the business of Elan Row hopefully David Moyes Declan Rice at your last game Declan Rice, can you hear me Declan Rice Go on do the bitch for them, Michael and Sony have always thought you're the boss Danny Inge, great fella Danny Inge by signing for Westam and Norris saves us saved with your charlison and Dan Jumeth Dan Jumeth is another one I didn't want to come to Evan because I thought I'd effect it more by not playing for Evan he was born ready he was born ready and Bourmouth, listen, you should have had a great team brilliant, brilliant we'll clap you off the pitch enjoy your go, no phones no phones in anyone's faces let us have a couple of goals and just go there you go boys, we're all over the globe we'll have chilling in our slide summer with our socks on with the socks on and the cans there you go, let us know what you think in the comments section below, can Evan and do it can Evan, do it please give us a video with thumbs up subscribe if you haven't and if you want to continue with our relegation battle therapy groups, lives everyday, people must off the TV premium in there, why wouldn't you the link is in the description I'll call dubby on the screen now, take it easy, bye


What advice would you give researchers to practice good science and avoid any kind of misconduct? So one thing is just to be honest with your work. So when you're doing the experiments, consider whether the results are true and valid and did the experiment work and did your controls work. Should an experiment be repeated? Is it properly analyzed? If those are all a guess, then you can use that data and consider it for part of your broader story. And then when it comes to publishing, it's really important to get the information for authors and ethics policies for whatever journal you're submitting to and reading those policies before you submit. Most journals follow similar guidelines, but everyone's a little bit different. And it's important to look at those guidelines well before you publish because as you prepare your paper, you want to make sure that your data is going into the right repositories.  And authorship guidelines are also important as to how they define authorship and whether they define contributors and authors or what should go in the acknowledgment section. Also issues with conflicts of interest, how do you report your interactions with companies or other private organizations that may be paying for your research. All those things should be taken into consideration early as opposed to after publication when the journal writes to you saying that something was not fulfilled properly.


It is now time, though, for Member Statements, and I recognize a member from Niagara Centre. Thank you, Speaker. Vaccines are the light at the end of a very long tunnel, but in my community in Niagara, the vaccine rollout has been rife with confusion due to the actions of this government. We still have no answers on the diverted shipment of 5,000 Moderna vaccines away from Niagara. I've heard from countless constituents from all age groups who were told conflicting information for the provincial booking portal and had difficulty booking their appointments. The province earmarked selected areas for priority access to COVID-19 vaccines, but people across the province were confused when some of the areas with higher case numbers were not selected. Niagara Region's Medical Officer of Health in an interview with TVO stated that they were not consulted and that they did not even receive detail on how the hotspots were selected. Ontario's own science table stated that they did not determine the actual postal codes that the government would ultimately prioritize in its vaccine strategy. Welland, a city in my riding, has the highest case count per capita in Niagara. The Mayor, the CAO, and the Fire Chief have written to the Premier, respectfully and urgently requesting additional shipments of vaccines to protect our community. Premier, it is imperative that you review the data, listen to the science, and get vaccines into Niagara and communicate clearly so that we can get them into the arms of our citizens as quickly as possible. Thank you very much. Member Statements, the Member for Stormont Dundas, Sotlanda. Thank you, Speaker. I rise today to shine a light on the goodwill and generosity shown by so many individuals and volunteer groups in my riding in Stormont Dundas in South Hungary. When we talk of Ontario spirit, they have shown its true meaning with their innovative ideas and they're giving their time to their communities. These acts of goodwill truly make a difference for so many, especially those who have difficulty getting out during this pandemic. The Optimist Club of Cornwall hosted a drive-thru parade in Halloween and Christmas. Cars lined up for over three hours to get through. The volunteer virus departments and community organizations organized Santa Claus parades and drive-thrus in Finch, Winchester and Chrysler. On an individual basis, neighbors picked up groceries and prescriptions for those who were unable to get out and shop. Others have shown face masks for friends and neighbors when they couldn't find them in the stores. There are also the volunteers who take our long-term care residents out for a stroll in their wheelchairs to the local for the Friday luncheon. And the same spirit, Mrs. Louise Valentine of Cornwall, notice of the Premier was constantly having to remove and replace this mask during his daily updates. So to make it easier, she made a number of mask holds and asked me to hand deliver them personally. Mr. Speaker, all over Ontario, but especially in Stormont Dundas in South Hungary, volunteers are stepping up to make life better for the communities. Thank you. Thank you. Member statements, the member for Kiwet and Nong. Hi, good morning, Speaker Miigwetch. Earlier this month, we learned the cost of policing at Land-Back Lane and Six Nations 3D territory. Between July 2020 and January 2021, Ontario spent $16 million on policing the Haudenosaunee people at Land-Back Lane. Speaker, that could have been all avoided. The nation has repeatedly called on Ontario to engage in a peace-proof process that would build a true nation-to-nation relationship. But this call was answered with racist and colonial violence from the OPP in Ontario. And the invitation to the Premier to visit Land-Back Lane has been unanswered since day one. Now the Haudenosaunee Confederacy at the Grand River has put in place a moratorium on development on the Haudenosaunee track. They have said, and I quote, our children are relying on us to protect them. This is a sacred responsibility that every Haudenosaunee person must uphold in no one, not Canada, not Ontario, and certainly no municipality has any right to interfere with our right to protect the lands and the waters that belong to our children. End quote. The council has said they understand that they share the lands and agreed to uphold the agreements between the people that live in peace, friendship, and trust. Mr. Speaker, it's time for Ontario to uphold the treaties. Miigwech. Thank you. Member Statements. The member for Prairie Salon must open. Thank you, Mr. Speaker. Yesterday I had the opportunity to join the Minister of Environment, Conservation and Parks, the Minister of Natural Resources and Forestry for an exciting virtual announcement. We announced that our government will fund 16 projects that resulted from the good work of the Muskoka Watershed Advisory Council, sorry, group. The government has entered into funding agreements with the district of Muskoka for 15 projects and the Town of Bracebridge for one project. We are investing $4.25 million in these projects to address three main issues. The first is a long-term goal of creating an integrated watershed management strategy for the Muskoka Watershed. The second is addressing flood medication in the shorter term, and the third is ensuring the health of the watershed and water quality. These projects have all come from the 19 recommendations made by the Muskoka Watershed Advisory Group. I want to take a moment again to thank the members of this group for their hard work and dedication to ensuring the health and the future of the Muskoka Watershed. Thankfully, this spring has not brought flooding or high water within the Muskoka Watershed, but after significant flooding three times in the past decade, water levels is an annual worry for many residents. That's why I'm pleased to see funding for a number of projects that float focus on flood mitigation. Many of these projects will involve a great deal of community involvement in cold saltations and in more formal roles within the community roundtable and task force. I hope that residents who are interested in protecting our beautiful lakes and rivers and wetlands will get involved with these projects. Thank you, Mr. Speaker. Thank you. Member Statements. The Member for Algoma, Manitoulou. Thank you, Speaker, and I've been in my seat over the last couple of months listening to this government make statements as far as misleading a disservice towards the opposition when they're raising the concerns that have been raised by many healthcare professionals across this province and community leaders in regards to bringing in paid sick days. We know that the federal sickness wage replacement program can only be accessed if a worker contracts COVID-19 and cannot be used to cover vaccination appointments. This is a fact. This government has been steadfast with their position claiming that the federal program answers all the needs of workers across this province. Recently, I've heard some backtracking coming from the House Leader, coming from the Minister of Health yesterday during an interview, coming from the Minister of Labor. I hope, and these are giving me signs that this government will be supporting our opposition motion that we will be bringing this afternoon in regards to focusing on bringing vaccination to essential workers and bringing in paid sick days to workers across this province this afternoon. Speaker, it is my hope that we will finally see this done by this government. We heard from many municipal leaders across this province. We have heard from the experts at the table what I'm asking this government and pleading with this government support the vaccination of a sectional workers and this afternoon you have the opportunity to bring in paid sick days. The House to make their comments through the chair. The next member's statement, the member for Ottawa State. Thank you very much, Mr. Speaker. And we all got a bit of an April surprise today. And I do have to say walking to Queen's Park this morning was really quite peaceful and beautiful. The place looked great. It was very healing. And nature always finds a way of giving us what we need. So we have to give nature what it needs. April 22nd is Earth Day. It's a celebration of the world's largest environmental movement. In fact, every day is Earth Day. And it's up to all of us together to protect our planet. Climate change is the greatest challenge for our generation. It is vital that we reduce our greenhouse gas emissions. And I was proud to be part of a government that understood this and introduced a plan to reduce our carbon footprint. This government has gone in the opposite direction. They entered the cap and trade program and put nothing in its place. They asked the environmental commissioner. They gutted conservation authorities. They passed legislation allowing a minister to pave over wetland to build a warehouse. And now they want to spend billions of dollars on a highway through protected lands that nobody wants. Speaker, good environmental policy is good economic policy. And to quote Ernest Hemingway, the earth is a fine place and it's worth fighting for. Thank you. Member Statements, the member for Kitchener Conestoga. Well, thank you very much, Mr. Speaker. And earlier this week, I was pleased to share some exciting news with Kitchener Conestoga and all of Waterloo Region. As of yesterday, everyone 40 and older is now eligible to get the AstraZeneca vaccine at participating pharmacies and through certain primary care physicians. COVID-19 has touched nearly every aspect of our lives and has presented challenges that no one could have predicted, Mr. Speaker. But we are getting closer to defeating this deadly virus. And the best way to do that is through vaccines. Let me be clear, all of the vaccines that Health Canada has approved for use in Ontario are safe and effective. Vaccines are the best way to protect your health and the health of those around you. And your first choice for a vaccine should be the first one that is available to you. There are over 50 pharmacies across Waterloo Region offering the AstraZeneca shot in Kitchener, Waterloo, Cambridge, Breslau, New Hamburg, Baden and Almyra. And if you live in Breslau or the surrounding area, this Saturday, Waterloo Region nurse practitioner led clinic is also holding a vaccination clinic at Breslau Commons Pharmacy. These nurses will be using their day off, Mr. Speaker, to get vaccines into as many arms as possible. So please, if you are eligible, visit ontario.ca slash pharmacy COVID vaccine to book your shot today. There is a light at the end of this very long tunnel, Speaker. And the more people who roll up their sleeves and get vaccinated, the sooner we can turn the corner. Thank you very much. Thank you. Member statements? Member for Toronto, Dan Ford. Speaker, thank you very much. I stand to recognize and thank Stuart Horkshank. Stuart is retired this month as the Executive Director of Don Valley Legal Community Legal Services. It started out as staff lawyer at East Toronto Community Legal Services, was Executive Director there and then Executive Director of the merged unit. I want to thank Stuart for all the work he did to protect tenants. For a number of years now, we've been dealing with renovations in my writing. People would find out that their landlord had sold their building and a new speculative landlord came in, intimidating people, harassing them in some instances, trying to bribe them to get them out because they knew they could make a fortune. Stuart made himself available to those tenants. If it was a Sunday morning meeting in somebody's living room, if it was a Tuesday evening in a parking lot behind the building, if it was in a room in a recreation centre. He came, he brought legal knowledge, he calmed people down, and he gave them the tools they needed to fight and protect themselves. He did an extraordinary job. And I want to thank him on behalf of my writing in front of Danforth for all he did to help protect people's homes and make their lives better. Thank you, Stuart Corkshank. Thank you. Member Statements, the Member for Eglinton Lawrence. Thank you, Speaker, and good morning. I rise today to thank this House for passing my motion 155 to establish a Holocaust Memorial by unanimous consent on Monday morning. Thank you. As we did not have the opportunity to debate this motion through the usual process, I just want to put a few words about the motion on record. Motion 155 states that the Legislative Assembly of Ontario will, in broad consultation and direct engagement with Ontario's Jewish community, establish a Holocaust Memorial on the grounds of the Assembly no later than 2025. Earlier this month we marked Yom HaShawa, the Holocaust Memorial Day, which is Israel's official day of commemoration for the 6 million who perished at the hands of the Nazi regime. Yom HaShawa, of course, is widely observed outside Israel as well, both by the Jewish community and others. In years past, these commemorations would happen in person, and I have attended many of them at synagogues in and near my riding, and I recall outdoor ceremonies at Mel Lastman Square as well. Last year and this year, many of those commemorations went virtual. But make no mistake, there will come a time when we can gather together in person once again to mark this tragedy, and I think a permanent memorial on the grounds of the Legislature will establish an excellent place for those gatherings to take place in the future, an excellent place for us to remember and reflect on the horrors of the past, and an excellent place to affirm never again. Member statements? The Member for Northumberland, Peterborough South. Thank you, Speaker. Speaker, it's an honour to rise in the house today. This past year has been incredibly difficult on the people of Ontario as governments across Canada and across the globe battle the COVID-19 pandemic to protect the health and well-being of their citizens. I want to take a special moment to highlight the remarkable work of our small businesses, Mr. Speaker. Speaker, these small businesses are the lifeblood, the backbone of our economy, the lifeblood of small communities like mine. But, Speaker, this pandemic has not been equitably felt by everyone, and it's been harder on many of our small businesses. Speaker, it's hard being a small business at the best of time. But I think to my friend George from Olympus Burger, who I spoke with the other night, who's pivoted, who's pivoted well with Takeout and who stood tall, offering gift cards to support other small businesses in our community. I think to Jess Dodd, new owner of the Beamish in Port Hope, in what should have been the most exciting summer of her life, but has instead been navigating the pandemic. Salon owners like Ben in Newcastle, whose stories will forever be etched in my mind, Reno, Ferrari and Cooper, Gina Boyd and Brighton, Sharp's Food Market in Campbellford, who had an army of volunteers delivering food to the most needy. Rob from Wrap It Up and Go in Norwood, who dealt with the tragic passing of his dad, but is pivoted to continue to grow that business. Mr. Speaker, this member statement isn't about the supports. This member statement is about them. The work that they do in our community, the selfless attitude, the get up and go attitude every day, the backbone of our economy. I thank you. I appreciate the work you're doing, and we all in this place and in this country owe you a debt of gratitude. Thank you, Speaker. Thank you very much. That concludes our member statements this morning.


Hey, it looks like our worshipful master YouTube has allowed me to post today and before you guys say to use Alternative platforms those are controlled by the same people So it's kind of like a trick to get certain influencers off YouTube thinking they have a chance If you guys have been wondering why I haven't posted the past few days You've been out of the loop. It's because YouTube has decided my burgers and fries day of eating video Violates their guidelines as usual. I posted Thursday at 11 a.m. And Instantly at 11 a.m. The video was taken down. I couldn't figure out why it would have been taken down So I just filed an appeal and later that night at 10 40 p.m The video was reinstated because they confirmed the content does not Violate the community guidelines now. I was hoping that was going to be the end of it But the next day at 10 30 a.m. That same video was taken down for violating the same Policies so after being up for about 12 hours overnight They took the video down again for the exact same reason so I filed another appeal and at 10 53 p.m The second night the video was again reinstated Determining it does not violate the guidelines this time I made sure to delete the video as soon as it was reinstated so they couldn't take it down again now My channel already has a warning for talking about some bills that were passed in New York State Congress last year So whenever they take down a video now, I instantly get a strike What that means is I can't post or comment for a week no videos no community page They could technically do this to me Every single day on every video I post for no reason and I can never post again I couldn't actually tell you guys anything on Friday or Saturday, and I was so disgusted I didn't bother on Sunday and we all know why this is happening. You know if it was the YouTube bot I think the video would have been taken down before it was live and There have been a few coincidences on my channel whenever I call out these rats for plagiarizing me Stealing my work to sell their agrochemical laden crap products You know the video gets suppressed only a portion of the views I normally get and the people don't suffer repercussions Despite them blatantly taking advantage of my hard work and what I've spent the majority of my adult life doing I made a video joking one of the Victoria's Secret models had Football player shoulders because her skeletal structure was wider than three of me and the next day I lost 5,000 subscribers. I mean, I don't think the joke was worth it and YouTube didn't like that apparently then when I spoke about my parents getting injected the video was taken down Immediately and after I appealed it they reinstated the video because all I did was read their own guidelines So there's two possibilities in my head here the more likely one being that I made the video on Wednesday talking about baby formula mentioning there are evil people that don't want you to know this I might have thrown the word YouTube in there The other possibility is the burgers and fries cheat meal video was getting too many views per hours for my channel Or I said some word in there. That's really bad in the algorithm So they had to put I don't know. I really don't know what doesn't make sense is Some YouTube algorithm bot taking down the same video twice You know, this seems like something a person would have to be involved with like they manually reinstated the video and then a second time I mean if it happened once you could say that but it didn't and we know the people in power control all forms of communication Anyone that gets popular on any social media platform and has continued growth has made a deal with the devil What discuss and discourages me is finding out that basically everything is fake Every influencer most businesses on a larger scale Everything that most people believe it's all fake everything on TV and at this point It's hard for me to want anything to do with it even pretend that some of it's real Now these scumbags spend so much time pretending to be good many illusions whether it's pushing a vegan diet while they eat sushi or a Green-fed corn of war diet while they feed their family grass-fed beef and then when someone good comes along and Points out their bullshit. They just close the curtains so they can continue to deceive and make money for themselves I mean, what am I supposed to talk about at this point? I can still talk about climate change being fake Right because that's not illegal yet Apparently cooking burgers and french fries is not allowed. Maybe we should try it again this week is Pointing out how the ingredients in baby formula shouldn't be fed to children a problem What about the fact that authority figures blatantly promote these types of products?  I mentioned earlier, so I Think that's gonna be it for today. It's early Monday morning, and I got to get down to work But you know overall, I'm just I'm just so I just discouraged with everything, you know, I've basically spent every waking moment, you know trying to help other people improve their nutrition educate them and You know, I do have a great amount of supporters and people that have helped me But yeah, yeah, I don't know how much I can take anymore. I'm just completely over everything So if you guys could please drop a like on the video share it I guess sign up for the newsletter on Frank dash the fire calm We have the 4th of July grilling pack available on Frankie's French meat And I'll talk about that tomorrow But if you want to get your order in time for 4th of July this weekend definitely place in order today


So, we were talking about products of quasi-projective varieties and we had this Sager embedding which was from Pn times Pm to Pn, some n tuple A0 to An B0 to Pm was sent to all the products. And now we had seen that this map is injective, the image is closed and we then wanted to say we identify the Pn times Pm with its image and so identify in P large n and also if x and y are quasi-projective and say x lies in Pn and y lies in Pm we identify x times y with image in the corresponding Pn. And we had seen that this makes some kind of sense, we had proven that if we define the product in this way it fulfills a universal property so if z is any variety then the morphisms from z to x times y precisely those maps such that the two, if I compose the map with the projection to x and with the projection to y both are morphisms. So now we will want to soon talk about these two properties that one defines using products which were separateness and completeness and they both have something to do with topology on the product so I have to tell you what the topology on the product is. I mean there's a risky topology on the product. So this is, I don't know, lemma so I want to tell you what are the closed subsets in Pn times Pm. It's actually quite simple so the closed subsets in Pn times Pm are the zero sets of sets of polynomials Cf which depend on the variables on Pn and on the variables on Pm and they are supposed to be homogeneous in these coordinates and homogeneous in these coordinates but not necessarily of the same degree so P homogeneous, pi homogeneous in the xi and the xj. So every monomial of f will be some, will be a product of a monomial in the x's and a monomial in the y's and the degree in the x's will always be the same and the degree in the y's will always be the same but these two numbers don't have to be equal. So now I want to prove that. So we have to remember that we had defined this Pn times Pm was identified in the yj. So we have to remember how everything was defined so we had identified Pn times Pm with its image in P large n and so that a subset here is closed means by definition that its image in Pn is closed or that it's the inverse you know as it's an isomorphism of this image that it's the inverse image of a closed subset in P large n. So let W in Pn times Pm be closed then W can be written as sigma to the minus 1 of a sigma was the sigma embedding for a in Pn closed. So a is a closed subset in Pn so we can you know it's a zero set of some polynomials. So in the variables on the P large n. So we can write, so a can be written as z of say f1 depending on the variable cij that we have on the large Pn until say some r depending on the zj's fi polynomials homogenous polynomials in the variables zij on Pn. So now how does one take the inverse image we compose we have to compose with the map you know we take the map sigma the inverse image on sigma so the inverse image of the zero set is the inverse image of is the zero set of the polynomials which we get by composing the fi with sigma and what does that mean we replace zi by xi times yj. So it follows that W is equal to z of f1 of xi times yj until fr of xi times yj. Now if I you know fi was maybe this is something called k as I have i and j so fk is a homogenous polynomials in this variable cij I replace zij by xi times yj so afterwards I get the polynomial which is by homogenous in xi and yj even of the same degree because it has the same degree each monomial has the same degree in xi as before it had in zij because I just in the xi's as it had before in the z variables and in the y variables also the same degree as it had before in the z variables so the fk of xi times yj are by homogenous in the xi and the yj even of the same degree but we don't care so conversely there would be several ways how to do it I'm not sure if this is you know assume that z is indeed the zero set of polynomials like that so f1 of xi so of all the xi's and all the xj's until maybe I call it g these are now different from before gl of all the xi's and all the yj's so these are all the variables and so assume w is of this form for gi gk by homogenous so in principle one could more or less make these steps backwards but I do it in a different way so I can also use these open cover by these sets ui uj I had before then if I take if I take say cannot call them i and j so fl and yeah okay so here we I mean it doesn't matter very much you have here x i and yj are i and j are somehow running through all of them but now I fix i and j and so then I say if I take a fi times fj of w intersect it with ui times uj remember that fi times fj is an isomorphism from from your i times uj to a n plus m well how do I get this here I put and ui is where the ith coordinate is nonzero and fi is obtained by dividing everybody by the ith coordinate and you know taking the corresponding coordinate so this is so if I take the image and this is just the zero set I claim of you know g1 of x0 and then I leave the ith coordinate I put equal to one and then the other variables run and y0 and the jth coordinate I put equal to one and the other coordinates are just you know the the map is just the zero set of you know so this is the locals so you know let's try to retrace this and the same for the other ones no okay so let's go to the if I ui on ui times uj is where here the ith and jth coordinate are not zero the fi i sends a vector with coordinates a0 to a n to the one where I you know where from a0 from a n where the ith coordinate is nonzero sends it to the locals where I did sense it to the quotient a0 divided by i1 a0 divided by ai until a n divided by ai and so you can see that if I do this if before it was a zero set of this polynomial then afterwards becomes this polynomial where I put this variable equal to one it's just what is and so we get this thing but this obviously is a closed subset in a n plus m so as fi times phi j is a isomorphism which from ui times uj to n plus m it follows that a w intersected this is closed so it follows w intersected ui times uj is closed in ui times uj and you know the ui times uj form an open cover of pn times pm so if that if the intersection of w with every open subset of a cover is closed then w is closed okay and so that's this statement so now we come to some of these these two properties so the first statement the first property that the variety can have and which actually according to our definition all varieties have is that they are separated and the statement there is that the diagonal is closed so I have to say what the diagonal is and then so definition so x should be a variety so the diagonal so delta x so is so is delta x which is just obviously the set you know all the points in the agony so the set of all points pp p and x which is some subset of the product of x with itself and you also have the diagonal morphism so I don't one doesn't need that very much but anyway the agonal morphism which is um say delta x from x so x times x p is mapped to pp so this is a morphism so I can write it as according to our definition as the identity on x and the identity on x no so it stands for p2p and so this is a morphism with image delta clearly the image is there and now so the first claim is that the diagonal is always closed in the in the product so lemma delta x is closed in x times x and then one also gets that delta x from x to delta x is an isomorphism but the last statement is kind of automatic once you know that this is closed and therefore it makes sense to talk about this so yeah so it's it seems maybe kind of obvious but it's actually needs a tiny trick um so which one to the main thing we want to show is that delta x in x times x is closed so every variety according to our definition is isomorphic to a locally closed sub variety of some projective space no so therefore we can we can replace x by an isomorphic variety and so we can assume that x is a locally closed sub variety so well and now we have to you know if you want to say that the diagonal is closed we have to somehow write down the equations no but we can make it a little bit easier by noticing that it's enough for us to see that um p in in projective space itself the diagonal is closed so we have that delta x is equal to the diagonal of p n intersected with x times x and so if we know that the diagonal in p n times in p n is closed then this will be closed in x times x so if delta p n is closed in p n times p n then it follows because x times x carries the induced apology that delta x is closed in x times x so we can assume x is p n so that and now we just have to see we just write down the equation for the diagonal so i cling that you just can write down the formula the diagonal in p n is the zero set of i take xi times yj minus xj times yi so where you know say x1 x0 to xn are coordinates on the first p1 first pn and y0 to yn are the coordinates on the second factor so where i and j run through all possibilities okay maybe i different from anyway it doesn't matter so i claim it's this so so in this case it's like it is indeed you can see that this is indeed a determinant of some of some minus so let's see so why is this if i lay take a point a0 to an and b0 to bn in pn times pn then this will lie in the zero set in the zero set here if you know all these terms are zero when i put ai and bj and what does it mean so if and only if if i take the matrix a0 to an p0 so so the entries here are zero a1 b1 and so on and so on a n bn if i take this matrix if this matrix has ranked one because you know the minus of the matrix are precisely ai bj minus bj bi aj so and the rank is one if and all two by two minus are zero because the the rank is at least one because these are non-zero vectors we know that so so this lies in the zero set if and only if the rank is one but that the rank is one means that these two vectors are put or proportional so this is if and only a0 to an is equal to b0 pn okay so therefore we have found indeed the equations for the diagonal in pn and so we know that the diagonal is closed and now the last statement is obvious so now we can talk about the morphism from x to delta x so as x is a variety i mean subjective morphism delta x will be a variety as you know because the map is subjective and it is clear that then delta x from x to large delta x is an isomorphism because obviously the projection to one of the factors is the inverse okay so this is this property so you know so you see it's not particularly difficult but we see that we have somehow explicitly used that somehow this is called a projective so that we can go into a projective space and in fact it is true that this property that the diagonal is closed does not hold in arbitrary generality so if you have a more if you look at schemes or something which would be a generalization of what we have here or abstract varieties which is something which locally looks like and fine like a in a fine variety then it will not be true in general that the diagonal is closed you know it's a property you can ask for you call the variety separated which we'll now say if this is true and we have just shown with this definition that quasi-projective varieties are always separated so I will just write this as a mark so then so that the fact that so the fact that delta x subset x times x is closed it's an important property and actually as I tried to mention before it somehow replaces for us the host of property okay so we know that algebraic varieties will never be housed off I mean except when they are points or something like that but this is something which implies some of the consequences of the host of property and so therefore we can sometimes argue as if we had the host of I mean we can get results which similar to what one gets for host of spaces so therefore it's a very useful property um and as I said you know a variety but is called separated if delta x subset x times x is closed and now according to if we take our definition of variety then all varieties are separated so that's not so exciting but as I said if one doesn't more generality than sep mark one cannot even read it at least I can't separate it so according to by dilemma with our definition all varieties are separated but there are more general definitions of varieties and then they aren't okay and so some simple consequences so um for instance so corollary if we have two morphisms from uh so let's say phi both phi and c from x to y the morphisms of varieties then the locals where these two morphisms are equal is closed in x so then the set of all p in x such that phi of p is equal to psi of p is closed in x so in particular if you have two morphisms which coincide on an open subset they are equal in a non-empty open subset for a non-empty open subset of x then it follows that phi is equal to psi because an open subset a non-empty open subset is always dense as x is irreducible so proof is quite simple so what is this set so this set that we have here is well the set where these two map to the same point so it's the inverse image of the diagonal by these two maps so this is phi comma is it like this okay so phi and c are morphisms so this map phi c no phi c is a map from x to y times y you know is a morphism delta y is closed in y times y so that's it follows that this is closed in x as the inverse image of a closed set by a morphism we can also see that the graph of a morphism is always closed it's like a some kind of closed graph theorem like you have in function analysis so the graph obviously is what it always is so let p from x to y the morphism of varieties so the graph of phi well it's just as you rule it's gamma phi which is the set of all pairs of a point at its image where p is a point in x so this is some subset of x times y and the claim is that corollary that first that the graph is closed in x times y and well it's also isomorphic to y so so I have that say p1 from so if I take p1 restricted to the graph by p1 and maybe p2 restricted to the graph from gamma phi to y is an isomorphism is it really obviously it's the other projection which is an isomorphism so p1 like this no it's kind of so if you have a a map you can write the graph and obviously if you project back to where you start it will be the trajectory projection and I claim it's an isomorphism but the main point is that this is closed and this is it follows again directly because it's again the inverse image of some closed set so I mean just gamma phi what is it it's the set of all points p phi of p with like this so I can take again a map like that I can take say I take for one thing the map phi and I take the identity on y so it's times identity on y with a minus one of the diagonal so this map phi times the identity on y is according to our definition a map from x times y to y times y which on the first coordinate is phi and on the second is y and then you know this says precisely that you know this is a set of all points p in x and you know q and y such that q is equal to phi phi of p and that's what it says so and this is so it's closed because this thing is a morphism and delta y is closed in y times y and that this is an isomorphism is kind of immediate we can take for instance we can see that the inverse of p1 restricted to gamma phi can be taken as so I say it's the identity on x comma phi from x to gamma phi this is anyway how you normally parametrize the graph no you on the first coordinate you you send a point to itself on the second you put it to the image and this should be an this isn't an isomorphism and the inverse is just a projection down to where you started with and you can see these these these are now both morphisms and they're obviously inverse to each other okay so this was this separatedness now we come to the other property that one can you know of morphisms and varieties that one can express best in terms of products which is completeness and whereas the separateness is replacing the house of property for us the completeness replaces compactness and you know as I had told you that we somehow want to view projective varieties as compactifications of you know a fine variety so we should have that the projective varieties are complete and this is what we want to prove and in fact according to our definition a variety is complete if and only if it is projective so if a variety is quasi projective and not projective or just a sub variety of refined space which is not a point then it will never be complete okay so let's try so completeness and this is a somewhat more subtle property so we should have that so we want that so as I said completeness replaces for us compactness and it's again a property one can describe in terms of of the of products so I will now introduce at the moment so first I start with something else definition so that you first you know so if you have a map say f x y of topological spaces and so such a map is called closed if well the image of any closed subset is closed closed in y for any or closed subsets z subset x okay so for instance yeah so somehow you know if it's continuous then the inverse image the inverse image of any closed subset is closed but now we have the image is closed okay and now completeness is a somewhat well strange statement so the statement is a variety x is called complete if whenever I do the when if the second projection x times y to y is a closed map for all varieties x for all varieties y okay so whenever I have a closed subset of x times y then the image in y will be closed and somehow you know that is not automatic it somehow is so let's see first some example but it's not true so for instance we have that a one is not complete and you can actually easy see that we'll see later that as I said if I find varieties if they are not points will never be complete but we just want to see it for a one a one is not complete so there are for instance we can take as the other one also a one so we take if you take z the zero set of x y minus one in a two which is the same as a one times a one so the graph of the real point somehow looks like this and we take the second projection this will map this thing the image will be everything without the old okay so p2 um from so p2 of c is equal to a one without zero of course this is a second projection this would be this is a set of all possibilities for the second coordinate and you know if you know if y is equal to zero there's no possibility to find an x so that you get one and so this is certainly not closed in a one it stands okay maybe let me see so you know one reason why completeness is very important is that you know it has some property that you also have when you have a compact variety so if you have a complete variety then any morphism starting from a complete variety has its image closed it's like if you have a compact topological space and you map to a house of space then the image of the compact space in the house of space would always be closed under any continuous map and it's the analog of that so this is a yeah maybe I call it proposition so if let x be a complete variety um and let um say phi from x to y be um be a morphism of varieties so then phi of x subset y is closed and the the proof is very simple because um we can use the graph of the map so let um we have gamma phi subset x times y is a closed subset and by definition the image of phi is the second projection of the graph um so thus if x is complete phi of x is closed okay so it's a very simple fact but it somehow shows again the analogy with the house of things so the image and so this actually does give us a new way to construct varieties so we don't have to explicitly write down the equations just if we have any morphism from a variety to another variety we always know that the image is a variety okay but now we want to show that um all projective varieties are complete that's not that's a theorem or projective varieties so first we make some so the main step is actually it will turn out that it's enough to prove it for to prove that pn is complete then it follows very easily for projective varieties and actually the main step is even slightly less than that so the main step is to prove one main step is that if I say for some reason I change p1 into p2 into p1 before because I find it easier so the first projection from pn times pm to pn is a closed map okay I mean in principle it has to prove that now as I have exchanged one and two that the projection the first projection from pn times x to pn is closed for any x but you know it's the start to do it if the okay there's a set this is once one has done this the rest is formal so you know then we have to do it so we take a closed subset here so it's an algebraic set so we can write x as a zero set of say some polynomials f1 of x y I now make my life even easier until f r of x y where here x is an abbreviation for x 0 for the variables x 0 to x n no on pn and y stands for y 0 to y m the coordinates on pm and then I want to make some assumption I claim that we can assume that all fi all these polynomials so it's a zero set where you know these fi are bi homogeneous and x and y because that was the lemma we proved in the beginning so we can so in particular each fi has a certain degree in x and a certain degree in y and I claim we can assume that all the fi have the same degree in y and assume all of i have the same degree which are called d in y why is that so assume I mean most of the time obviously it will not be the case so one of them will have the largest degree one of the degrees one of the degrees one of these polynomials fi will have the largest of these degrees in y you know I take the maximum the other ones have possibly smaller degrees and then I claim I can replace them by other polynomials so that the zero set is the same and they have the same degree so if fj has a lower degree I mean then the maximum of these then we can replace it by other polynomials namely I say so assume the degree is the maximum degree minus e then I replace it by y zero to the e times fj y1 to the e times fj and so on until y n to the e times fj so if this had degree e less than the maximum degree now all of them have this maximum degree d and the zero set is the same as before because what is the zero set of all these polynomials it is the the the union of the zero set of the fj's and the common zero set of all these but the common zero set of all these is empty so it is just the same as the zero set of fj which has the same zero so by possibly making the number of polynomials we are looking at here much larger I can assume that all of them have the same degree in y so okay until now we are only we are still kind of so somehow I will eventually have to prove that the image that the second projection of x the first projection of x in p n is closed so you know that it's closed means that it's a zero set of some polynomial so somehow I have to create these polynomials out of nothing and so you can somehow see that this cannot be so easy because we don't see them so we somehow and that's why it's slightly tricky so we have to go about it slightly slowly so we fix a point p in p n and we want to investigate what is the condition for p to lie in the image so then p will lie in the first projection of x but if and only if there is a point in x whose first coordinate is p so that means that if I write it now like this if I take the zero set of these same polynomials as before f1 until whatever and now I just write f1 of p y as you know where I put instead of x the coordinates of p so there's comma so one does it lie in the image if there is this is the set of all points in x whose first coordinate is p so this this set must be non-empt this is some subset in in p m and we have to see that this is a closed condition so now we use the projective nulsternsatz so we are using a somewhat big theorem so what does the projective nulsternsatz say it says that the zero set is the empty set if the corresponding zero set of some polynomials is the empty set if the ideal generated by these polynomials contains all monomials of a certain of some degree for some degree it contains all monomials of that degree okay so now we have the opposite statement so so this is equivalent but this is equivalent to so for all s bigger than zero if I take the ideal generated by these polynomials these are now polynomials in the y variables this ideal does not contain all monomials of degree s in the variable okay in this condition for given s I may be called star s because I will have to reformulate a few times so now it's clear we have said that the degree in the y variable of all these polynomials is at least y no it is precisely y so in particular we find that it's impossible so this cannot be true so this so the condition star s is obvious trivial so it's trivial 2 for s smaller than d because there's actually this thing does not contain any monomial of degree smaller than d so in particular it certainly does not contain all monomials of degree d so thus therefore it's enough to show for all s bigger than d bigger equal to d if I let if I look at the set of all points p in p n such that p satisfies this condition so that the ideal generated by these things does not contain all monomials of degree s this set is closed in p n so because then so for each s this is a closed set because then so I don't know if I give this some name I call this say a s well maybe not a s whatever x s because then we have that the first projection of x is just intersection of all s bigger equal than d of x s no because the statement is that the image consists of all points p in p n such that this condition star s is fulfilled for p for all s bigger equal to d and so if for any given s this is a closed subset then the image is is the intersection of these closed subs and therefore is closed so so we have to show this statement so we have to fix an s bigger than d and we have to show that this is closed in p n and for this we actually do have to now give some equations for it and we will do this by some linear algebra so we fix s bigger equal to d and now we just you know write down what we have so we denote so I will quite a lot of notation so anyway denote say d by mi of y this is so mi depending on the y so y always stands for y 0 to y m y n so to y m and this so this is from i from one so m plus s choose m these are the monomials in the y variables of degree s anyway so there's certain we have so many we have a number of these monomials can just write them down doesn't matter what the number is there's some number here and we so denote so first this like this then we also denote by say n whatever j of y and now I don't nearly need to again j runs to something the the monomials in y of degree s minus d okay if you can figure out how many these are so these are always i somehow number them in any random way no just have one for every one again in any order so then we find so the elements of degree s in this ideal that we have here f1 of py f r of py are all those which are obtained you know all these have degree d in the y variable so they are just obtained by multiplying these elements by all monomials of degree s minus d they are just they are precisely so they are the linear span so the span in terms of vector spaces of of the set of all products n i of maybe n well now I call them n i I mean anyway sounds over all possibilities n i of y times f j of py so for all i and j so i this was would be m plus s minus d choose if I want m so i from zero this thing j from zero to r so I take these are we have these elements they spend some suck vector space and that's up to vector space are all the elements of degree s which lie in this idea okay and we somehow want to see that this does not contain all monomials of degree s so it's a question about the dimension of the span so I write down so denote again the product all the products I take all the products n i of y times this thing but I actually take it now with the variable x instead of the p so denote so I take all of them so by somehow g k of x y so there's some kind of way how I renumber them know so that I have some case which run from somewhere somewhere in you know from this is from one here so from one to the product of these two numbers and I have all these polynomials in x and y okay I make quite some story you know k equals one say to t where t is the product of these two and this again in any order so now I have somehow made these complicated things so I can reformulate the condition star s now this is equivalent to well okay t would turn out to be by what I just said I mean somehow irrelevant what it is it's just some number but it will be equal to m plus s minus t choose m times r no it's just all possibilities of multiplying them so then this condition is equivalent to the following if I take these polynomials that I have here all the polynomials which are obtained as products so and I now put back the p for the x so g k of p comma x for one smaller equal to k smaller equal to t so this finally many polynomials do not spend the whole space of polynomials of degree s or there was a y here not x because according to what I said the elements of degree s in this ideal are equal to the linear span of these elements here and the statement that this does not contain all monomials of degree s just says that this linear span is not equal to the set of all polynomials of degree s okay but now you know until now it looks like I have really done nothing but now the point is we can express this in terms of looking at the rank of a matrix so let's see maybe I can so we can write if I take this g k of x y so I'm back to the polynomials I don't put the p I can write this as a sum over all I now I wear a i j a i k of x times mi of y mi are the the monomials of degree s so this is would be i from 1 to m plus s to s or m whatever it's the same no because where this a i k from x is homogeneous in the x variables you can easily see that so I mean it just means I have a polynomial in x and y which is homogeneous of degree s in the y variables then I can and it's also homogeneous of some degree in x variables and I can write it as a sum over you know some these monomials in y times some homogeneous polynomial in x okay so I have this matrix of polynomials but now you see this somehow gives us you know we have a this linear map so then the claim is the dimension of the linear span of say this g k p x which is what we were wanting to know k equals 1 to t is just the rank this matrix I don't know which way we're out of no goes t times m plus s choose m matrix a i k of x no because here we have a you have a basis of this vector space of monomials in y and you want to know you know we write down these things as linear combinations of you know if I put here p this is just a number so you know if I put the p so here p so if I put p this is just a number and so then if I write the you know some vectors in terms of a basis via elements of a matrix then the dimension of the span of the vectors is equal to the rank of the matrix okay and so therefore we see that the condition that the linear span of these things is not equal to everything is equivalent to the fact that this matrix does not have the full rank so so thus this condition star s is equivalent to that the rank of the matrix a i k of p where you know i went from 1 to m plus s choose m and k from 1 to t that this rank is strictly smaller than m plus s choose s no then the dimension of the vector space in which this thing here lives and so this means that if I look at the minus of this matrix of this thing they should be zero so so now I can wipe out maybe the condition so thus the set of all points p in p n p n such that this condition holds for p you know the condition was this is equal to the zero set of all m plus s another word s but if I wrote m although it's the same when we write all the same m times s times m plus s m minus of this matrix so I have this matrix so the minus means I have the determinants of the sub matrices no and so it's a matrix which has different lengths of column and so anyway so it should not have the it does not have the full rank it doesn't have the same rank as the dimension it has in one direction if these minus are all zero well and you know this is a matrix with entries polynomials in the x variables so the minus are the determinants of these sub matrices they are polynomials in the x variables so it is a zero set of some polynomials so thus we have that the set holds for p p is a closed subset is closed in p n and as we had seen we have that the image p one of x is the intersection of all of these for all possible s so p one of x is closed in p n okay and so this proves the first statement that if I take the map p n the first projection p one p n times p m so p n is a closed map okay so now we want to see how one deduces the statement from that to prove that all projective varieties are complete and we haven't even quite shown that p n is complete we have just shown this statement that I just wrote so let's now show first that p n is complete so that we do the general case so first show p n is complete so that the statement is that if I take you know p n times y and project to y no no then this is a closed map so let's see so let for any variety so let y be a variety so we know that every variety is isomorphic to a quasi-projective variety no so it's even if it's either locally closed in in a n or p n for some n and then but if it's a locally closed in a n we can it has isomorphic to to something in p n so we can assume that y is quasi-projective so it lies in some p m and is locally closed there now we take z a closed subset of which we're going to go so a closed subset of p n times y we have to show that the second projection to y of that is closed in y okay so we certainly can take so let z bar be the closure of z in p n times p m that we can certainly do we can close it so then by what we just proved we know that if we take the second projection of z bar bar this is closed in p m because we know that the you know the projection from p n times p m to one of its vectors is a closed map so now I claim that what we want follows from that we have to show that the second projection of z is closed in y so let's see so what is the second projection of z well I claim this is just a section projection of z of this z bar intersected p n times y now because z is closed here so it is it's therefore the intersection of p n times y with its closure in p n times p m it's a section of so of p n times y with its closure here so it's just this z is just equal to this but you know if we project here you know what happens is just by definition this is just the second projection of z intersected with y you know and this says that it's intersection of y with a closed subset so it's closed okay and you know the now we come to the really general now we want to show that any projective variety is complete finally so finally let x subset p n be a closed sub variety and let z contained in x times y be closed you know x times y is closed in p n times y you know so therefore it follows that z is also closed in p n times y because it's closed in something closed and now I can just apply the previous thing if something closed in p n times y then the second projection to y is closed in y therefore by previous step p2 of z is closed in y okay and so with all this we finally have shown that projective varieties are complete so as you see it was a somewhat tricky thing these steps you know it's also a little bit with trick but it's kind of formal you know you have to somehow feel it's a bit like cheating but the most of the work is is done here to actually write down the equations in the case that we are dealing with p n and p m and you know and was and you can see that the proof is sufficiently complicated that we have no idea what the equations are actually you know which is also how it should be because you know we don't know where they come from so therefore the proof has to be difficult okay maybe I stop here and then the next time we will give some some applications of this and then we start with something new okay