Good afternoon. I'm very pleased to welcome you all to this IIEA webinar today. My name is David O'Sullivan, I'm the Director General of the Institute and I have the honor to be moderating this event. This event is part of our global Europe project, which is supported by the Department of Foreign Affairs. This project aims to analyze and communicate to the wider public the debate on the future of Europe, the EU's role in the world and Ireland's role in the multilateral order theme very important to the discussion we're going to have today. I'm extremely delighted to be joined today by Marija Petrinovich Gurish, the Secretary General of the Council of Europe, and Simon Kovni, Minister for Foreign Affairs and Defense. The minister will be the incoming chair of the Committee of Ministers of the Council of Europe in the second half of this year. Our speakers will address us for about 15 minutes or so, and we will then move into a question and answer session with our audience. I'd like to remind you that today's discussion is on the record. You'll be able to join the discussion using the Q&A function on Zoom, which you should see on your screen and we should all be fairly familiar with that by now. Please feel free to send your questions in throughout the session as they occur to you, and we will come back to them once we move to the question and answer session. And please feel free to join the discussion on Twitter using the hashtag IIEA. We're also live streaming this morning this discussion in so very warm welcome to all of you tuning in via YouTube. Having finished the housekeeping, let me now introduce Minister Kovni and give him the floor. Simon Kovni TD is Ireland's Minister for Foreign Affairs and Defense. From May to November, he will chair the Committee of Ministers of the Council of Europe as part of the Irish presidency. Mr Kovni previously held the position of Thonishta, Deputy Prime Minister from 2017 to 2020, and a number of other senior ministers, ministries. He is a former MEP for Ireland South, and he is a TD for Cork Central Minister, the floor is yours. Thank you very much, David, and thanks for the introduction. In you and in Maria, I have the honor of speaking today with the former Secretary General of the European Commission and the current Secretary General of the Council of Europe. But let me start by quoting a third Secretary General from another vital organization of which Ireland is a proud member. Once at the height of the Cold War, following a debate between the two blocks at the UN, reporters surrounded Secretary General Hag Hammershead, pressing for a comment. And the Swedish diplomat was evasive, as Secretary Generals can sometimes be. At length, the reporters grew exasperated at his replies. Could you say at least one of them demanded whether the compass points left or right, east or west, and Hammershead paused before replying, it points forward. Today, amidst war in Ukraine, our continent stands at a crossroads. At such times, we should hold our compass close and orientate ourselves by first principles. And Ireland's compass is the multilateral system that we've helped to build over many years. Our first principles are democracy, human rights and the rule of law. Principles first codified on this continent by the Council of Europe and promoted and protected by it still. 50 years ago, today, Irish people voted to join what has now become the European Union. For a quarter of a century before we did that, we lived and shaped European values. In London, in 1949, we were amongst the 10 original signatories to the statute that created the Council of Europe and the European Convention and Court of Human Rights. It was an Ireland's initiative that a commitment to the pursuit of peace was added to the preamble to the statute. And it's that pursuit of peace and accountability for its violation, which occupies so many people's minds today. Secretary-General Buryk met my Ukrainian counterpart Dmitry Kuleba in Kiev yesterday. Dmitry was an ambassador to the Council and is a firm believer in its values. Last month, I, too, visited Kiev and Bucha at his invitation. Together we saw the truth of what the Kremlin still calls its special operation. Through rubble streets, we walked a flattened, blackened city. We stood by trenches in which hundreds of innocents lay buried and listened to those who had survived the onslaught. A few months ago, the idea of such carnage unfolding on our continent seemed not only unthinkable, but virtually impossible, but no longer. And the world has changed utterly, and we are at a turning point, whether we like it or not. On the 20th of May, at this point of profound challenge for our continent, Ireland takes the helm of the Council of Europe's Committee of Ministers for a seventh time. In the same month, we celebrate our 50th anniversary as an EU member. In the same year, we serve on the UN Security Council. We are honored to steer an institution that inspires, not with examples of power, but with the power of example. The Council and Court have long served as the conscience of Europe. And across our six month presidency term, our goal, working with the Secretary General and her team, is to reaffirm that conscience. As presidency, we will strive to serve the Council as an adjusts to the expulsion of one of its largest members, and as it refocuses its resources to respond to the plight of another. Within this context, we will pursue three clear complementary priorities. Let me consider them in turn. The first is our founding freedoms. We will use our mandate to reaffirm the Council's founding freedoms, renewing our focus on the protection of vulnerable citizens through the effective functioning of the European Court of Human Rights. The Court is where the conscience of Europe truly lies. We were the first state to accept its jurisdiction. And we've always abided by it. Through the decades we've had our share of judgments. Some were historic. Several were at their time contentious, but all were respected. Accepting our state was in the wrong wasn't always easy for governments. It's not easy for others today. But it is always right. The law that was ignored is a human right infringed. That's how it works. And if we're selective in applying the rule of law, rest assured before long lawlessness will become the rule. By protecting individuals rights, the judgments made by the court, the standard set by the Council spurred our state to reform and our society to evolve. We took only at the case of Senator David Norris took to the European Court of Human Rights in 1988, which resulted in the decriminalization of homosexuality in Ireland. The joy our nation shared when the marriage equality referendum passed so resoundingly in 2016 can be traced to that Strasbourg courtroom to the bravery of Senator Norris to the barrister who represented him, our future president Mary Robinson and the wisdom of the judges on that bench to the principles of the convention, they are bound to interpret, and we are committed to uphold. I'm here today to speak of Strasbourg, not Stormont, but of course the convention forms a foundational part of the Good Friday Agreement also. In the wake of the troubles the human rights obligations it guarantees were crucial in building and bolstering public confidence in policing and political structures across Northern Ireland. They remain so today. Under the Good Friday Agreement, the protections guaranteed the people of Northern Ireland by the European Convention and Court of Human Rights must be retained and cannot be diluted in any way. The second area that we're going to focus on, we call here our voices. In January, we marked the centenary of our state's independence. We understand how hard the struggle for democracy can be and how being an act rather than a state. It must be renewed by each successive generation. Our second presidency priority is rooted in an abiding belief in the power of deliberative democracy and the necessity of youth participation. The Council of Europe has long held in promoting the rights of children and youth pioneering vital training and inclusion programs across the continent is essential. Through our term, Ireland will draw on its expertise to engage with and listen to young voices. They are the future of our democracies after all. In the face of rising illiberalization, we will draw from our national experience, above all with citizens assemblies to promote participatory, participatory democracy. And we will look to learn ourselves from others, recognizing the Council of Europe's critical role in setting standards that steers along the path of progress. The third area is Faulcia and those of you who are Irish will know what that means, and those of you who don't will soon know what it means. It means welcome. Under the Irish term Faulcia, our final priority will draw upon the changes our society has undergone since Ireland last held a presidency in 2000. As we seek to foster a Europe of welcome inclusion and diversity. We now face the largest refugee crisis Europe has witnessed since the Second World War. Already more people have fled Ukraine since February, then live in our entire state. The Council and its conventions affirm why we need to play our part in responding. But for the Irish people, this goes beyond politics or principle. Our collective cultural memory understands what it means to be forced from home. To arrive in distant lands carrying little more than the clothes on your back. For us then Faulcia is less a greeting, perhaps a creed. Already we've welcomed over 30,000 Ukrainians to our shores. We'll continue I know for however long it is needed. But we recognize also the great challenges that those tremendous flows of vulnerable fragile people present to politics across Europe. And the need for states to work with and learn from each other to protect all those who have shot who have sought shelter is an essential part in what we need we all need to do together. To further these three priorities, Ireland will make additional voluntary contributions of almost one million euros to the Council this year. Across our six month term we will chair more than a dozen meetings of the committee of ministers in Strasbourg will convene more than 30 conferences and seminars across Ireland. We'll invite the continent's justice ministers to double to strengthen European standards and combating domestic sexual and gender based violence. We will gather scholars and policymakers to go away to chart a path to enforce the European Convention on human rights in areas of protracted conflict. We will do all this and more conscious of the great challenges facing our country our continent and our Council of Europe today. Ireland would fully dedicated to addressing them together and determine that over our busy six month presidency term our compass, Europe's conscience shall point not left or right or east or west, but forward in the right direction. I look forward to your to your questions and your comments and more importantly, I look forward to the next six months when I hope Ireland will be able to provide both energy and leadership to the work of the Council of Europe at a time when it's more needed than ever. Thank you very much Minister for setting out those clear Irish priorities for the presidency which are directly relevant to the tragic situation in which Europe finds itself today. I will now like to introduce our next speaker Marija, but you know which Bush, who is the Secretary General of the Council of Europe. Prior to holding this post she served as Deputy Prime Minister and Minister for Foreign and European Affairs of Croatia. And in that capacity, she chaired in her turn the Committee of Ministers of the Council of Europe during the Croatian presidency from May to November 2018. General the floor is yours. Thank you, Chair, the Minister, the assignment. It's really a great pleasure to have the conversation as island prepares for the presidency of the Council of Europe's Committee of Ministers. It was also good to be in Dublin. Last week, I had meetings with President Higgins, with you and other ministers, and with members of the Irish delegation to the Parliamentary Assembly, and all of these were positive and upbeat. It's clear that Ireland is actually is important to this presidency and that everyone is determined it should be should make an impact at this very crucial time for Europe. This approach is long standing. It is of course, one of the Council of Europe's 10 founding members as it was already said, and throughout the organization 73 years history. It has been an advocate for multilateralism, and that we at the Council of Europe represent today plays an active role in protecting and promoting human rights democracy and the rule of law across our common legal space. In the good times, and in the more challenging moments, and certainly we are in one of those moments now. Russia's ongoing aggression on in Ukraine continues to inflict a terrible harm, as I could have witnessed yesterday, stories of rape, torture and indiscriminating killing are shocking and appalling. And our hearts are filled with sorrow for the Ukrainian people. Few of us imagined that in the year 2022, a Council of Europe member state would inflict such violent injustice on another. So our committee of ministers was right to expel the Russian Federation from our organization last March. It was clear from the beginning that the aggression against Ukraine was a flagrant violation of our statute, a clear fracture of the values that we exist to uphold, and something that we cannot accept from a member state. Our immediate priority is to do what we can to support Ukraine. This includes amending our action plan of measures on the grounds there to take account of the current realities. And I have established an expert advisory group to support the Office of the Prosecutor General, providing strategic advice as part of the process of investigating gross human rights violations in the context of the current aggression. And we will be ready to provide what help we can within the terms of our mandates. When the violence is over, a 50 million euro package has been put together for that purpose. Next week, our foreign ministers will gather in Turin for a ministerial session. And for other things, they will have the opportunity to endorse our changes to this action plan, and to again recognize that multilateralism remains the only means to build a future of peace security and cohesion in Europe. They will be affirming their determination to ensure that the Council of Europe remains the benchmark for human rights democracy and the rule of law in Europe. Throughout the following months, there will be no doubt further developments in Ukraine, where these concerns are overall, we will be sure to respond. The first presidency will play a central role in ensuring that we move forward, United. This was not an issue that you would have anticipated just a few months ago, the Foreign Minister, but it is the harsh reality of where we are today. I'm impressed by the way in which you have placed, placed it at the heart of the ambitious priorities that you have set out. You are right to place a focus on our pounding freedoms, reinforcing human rights and the protection of civilians in Europe, and right also to recognize the spectrum of what this means, ensuring from protecting individuals rights in conflict areas like Ukraine, but also in frozen conflicts throughout our continent, through the tackling that the violence against women and girls, which continue to blight the lives of so many people who are forced to live in fear. I'm also very interested by your choice of promoting participatory democracy and youth engagement as a theme. Democracy is in distress on our continent and beyond. So it is critically important that we discuss ways in which we can strengthen people's involvement and their attachment to democratic processes. This I think we will certainly benefit from insights from the Irish experience, citizens assemblies, citizens engagement, and the promotion of the rights of children and youth. In this, I'm pleased that you intend to advance the implementation of the Council of Europe's strategy for the rights of the child, and interested by the Global Forum on democratic missions of higher education, which will be hosted in Dublin City University. Taken together these initiatives and others will give all of us a greater greater understanding of how we can invest in our democratic future. Finally, your decision to prioritize a Europe of welcome inclusion and diversity is also highly relevant to the times. In recent weeks, millions of Ukrainians have fled their homes, seeking safety in other European countries. And many of those countries are going to great lengths to accommodate and support them. This is good, but not always easy. And we are working with our member states to provide the help and support that they need to uphold the human rights of those who arrive. The issue will be live through throughout the Irish presidency. But so too will other matters that you have flagged, among them combating racism, LGBTI plus rights and human rights and the environment on which the committee of ministers is scheduled to adopt a recommendation. A process in which islands guidance will be crucial. It is important to say that all of these key issues fit the strategic framework of the Council of Europe. This lays out the organization's priorities for a four year period. The themes and innovative events that you propose will provide further drive and energy in achieving our objectives. And I would like to finish by saying how greatly I appreciate this event too. It is a real sign of your commitment, of course, but it also helps get the message out there about what the Council of Europe will do under the Irish presidency and beyond. A future defined by human rights democracy and the rule of law is the best future for Europe. Recent events have demonstrated the terrible alternative, and we should say that openly and loudly for all to hear. So I look forward to working with you and wish the Irish presidency every success. And I look forward also to the questions that we may have from the audience. Thank you. Thank you very much, Secretary General, for that presentation of the work of the of the Council. I'd like to ask you both, maybe just following on from the three priorities which the Minister has outlined and which you have commented upon Secretary General. Quickly to run through them again on on taking the question of human rights. Could you describe Secretary General in a little bit more detail how the Council of Europe will address the issue of human rights violations in the context of the war in Ukraine and I think we're all deeply concerned about what is happening. On the first priority of the presidency. Can you explain the sort of mechanics of how it works from from the Council of Europe side and perhaps then minister you might like to comment on how the Irish presidency would want to take that work forward. That was actually one of the first things that I was asked by the Prosecutor General Madame Benedict of the Prosecutor General of Ukraine, while the war started she was already early March calling by video conference and asking for expert assistance advising on how to collect how to document and how to keep the the evidence of the gross human rights violations so immediately. My director general was to meet was meeting with the with the Prosecutor General, and since then we have an expert group advising her on these important matters. I know that as unfortunately the situation on the ground is appalling, shocking and apparently also our human rights commissioner was there for the whole week. Last week and she will come up with some in some future for the report on on on these human rights violations that she have heard the testimonies from. There are already evidences that terrible things have happened and still are happening. So we will certainly need to engage more with Ukraine in that respect, so I also called we had just last week at the conference of prosecutors general in under the Italian presidency in Palermo and Prosecutor General when they did to address the conference. So she was calling actually for to all prosecutors general to assist her in the best possible way in order to do that job because I think what is very important. While these gross human rights violations happen that the perpetrators can be traced and brought to justice, be it the national judicial instances or or international tribunals. So I think what what we can best do, while we see these horrible crimes happening is to assist that to those commits these crimes are should be brought to justice. This is the very immediate assistance that Council of Europe could do and I think in the future, we can see because I know that I CC the International Criminal Court has already been in Ukraine and it can in its mandate is to prosecute certain crimes, which is also to a little bit enlarge the some crimes like crime of aggression, but for that, I think international instances yet not have not yet announced or how to do that but that might also come up in the future of a parliamentary assembly has addressed that in one resolution in its meeting last last month, and now it is it will be for the member states but also for the larger international community to reflect what sort of international instance would it be only I CC or also to to start also working on on this issue, while we have evidently gross human rights violations happening in Ukraine. Yeah, I mean just on that issue. So, I mean, quite a number of the countries on the Council of Europe are already involved with working with the, with the International Criminal Court, in terms of funding the gathering of evidence, documenting evidence. I think he's actually the day before I was there Korean can was there with the chief prosecutor for the I CC. We announced funding of 3 million euros, while we were there. Actually, not just for their work in Ukraine but, but certainly for that too. We can ensure that, even though the prospect of taking a case in front of a judge of war crimes and crimes against humanity may seem like a remote prospect right now to actually build up the evidence base and to have the evidence file gathered and in this case is is I think really important. And so the chief prosecutor in in Kiev has been working with cream can and his team, and, you know, Ireland, along with another 39 countries I think referred the case to the I CC initially so I'm certainly going to be very interested in seeing how we can make a consensus across the the countries within the the Council of Europe around supporting that process, and actually then looking at how, and what kind of structure may be suitable for the crimes that are being committed that are outside of the jurisdiction of the I CC. I believe that the I CC can do and should be doing because it's their role. Taking cases against countries that are being accused of crimes against humanity or, or war crimes. What they can do is, is take a case against a country for aggression, which, which would need a, an independent tribunal type structure to be set up to take those cases around and I think that is the kind of the kind of structure and debate and consideration of the Council of Europe could look at I mean it is after all an organization who's raised on Detra is about protecting rights, protecting vulnerable people from aggression and holding states and state actors to account. And if there was ever a, an example of, of all of that coming together in terms of accountability and responsibility. I think the war in Ukraine is a very good example of that so you know I think that is something that certainly I'd like to to discuss with others and with with Maria as well, because I know there's been some discussion on it, but there are dangers here that a different entities will look at sort of duplicating effort. And, and one of the things I think we need to do in terms of having international hearings around the crime of aggression in terms of the Waging War against Ukraine and its people is that we've got to try to build as much consensus as we possibly can around how that might be done, what would be credible and fair in terms of a hearing system and consistent with international law. We know the ICC can't do that particular piece of work. So it's perhaps something that we could, we could get, get the legal team that advise the Council of Europe to look at and try to build some consensus around over the next six months. Thank you both very much. And we have a question from Andrew Ford who's a member of the IAEA's Young Professional Network, a question for Secretary General Burish, and the question is, are there opportunities for the Council of Europe to work with democratic forces in Russia and perhaps Belarus, as was committed to by the Committee of Ministers last month and how might that engagement, how, what is that engagement likely to look like. Thank you. Thank you very much for that because coming back from Kiev just this morning, actually one cannot then think about the whole why this terrible war happened and how we can engage with those who are still existing forces in Russian Federation in Belarus. And actually when after the decision of expulsion and the only right one at that moment has been taken by the Committee of Ministers, we immediately say we need to think about engaging with the democratic forces. Now, the situation is a little bit different for between the situation in Russian Federation at the moment and Belarus. Belarus has not been or is not a member of the Council of Europe for obvious reasons it still has capital punishment and although we try to work for over the years with the civil society and also with the government in order to come to a point where they will abolish the debt penalty and also come more as more as democratic society which it didn't happen probably the opposite on their on that side but so far we couldn't engage with directly with the opposition. Now the Committee of Ministers has taken decision that for the time being because Belarus being also used used as a territory from which Ukraine was attacked and also allegedly some some other things were happening for from from from Belarus. The Committee of Ministers decided to suspend all relations that we already had a very very low level or almost non existing and formally turn our relation to democratic opposition which is mostly outside of the Belarus. So I'm looking forward I know that the Parliamentary Assembly is planning to invite and also probably the Committee of Ministers Madame Tihanuskaya, with whom I was in contact already in one of the future sessions so that would be certainly a way forward. Now for Russian Federation is is a bit more complicated in the sense that Russian Federation seen it since the start of the aggression on Ukraine has harshened the the human rights and also some legislation in internal legislation in Russian Federation. For instance, at this moment, Council of Europe is treated as a foreign agent, according to their law, because if we were to fund a euro to any of human of NGOs or human rights defenders. They are treated as a foreign agent and be as provider of such assistance would be the one. So we, the Committee of Ministers decided to reflect a little bit how to construct the this type of relation with Russian democratic forces without putting them in a danger of possibly being put in jail or have suffered otherwise, but we have clear, clear will and clear strong commitment to work with forces in both countries in Belarus and in Russian Federation, because what we believe is that certainly we need to strengthen and work with those who are willing to to engage with the values and standards that we promote so there will be I'm sure during the Irish presidency, a possibility to have exchange with these people we already are engaged with some in some dialogue with both the Russian and Belarusian democratic forces and I hope that during the Irish presidency we will shape more concretely the way and strengthen these relations. Thank you minister this is related of course to your very much your second priority for the presidency would you like to, would you like to comment. I mean, look we've we've within the European Union there's already there's already quite a lot of dialogue, particularly with the with with opposition forces in Belarus for quite some time. I mean, every, you know, a couple of times a year we have breakfast briefings and meetings before foreign affairs Council meetings with with civil society groups with opposition groups and so on, in particular on the obviously, since the war in Ukraine began that has kind of dominated everything, but but I'd be really interested to see how we can use the the Council Europe structures to to create a platform for civil society for democratic democratic leaders and forces from either Russia or or Belarus to actually help them get their message across and to to ensure that that they have as much credibility as possible because you know in some ways this war. While there hasn't been a lot of focus on Belarus is is impacting hugely on on Belarus's fight for democratic change as well. So I think one of the things we can do is we can invite briefers from civil society organizations and when appropriate. We can invite Russian politicians as well to to ensure that that everybody understands that there are other perspectives in both Russia and in Belarus, even if they are at times brutally put down or threatened or intimidated by long prison sentences and or worse. They are brave people that are trying to bring about legitimate democratic change for the better, often inspired by the European Union and indeed other countries in the Council of Europe. And, and I think we need to, we need to look for ways in which we can give them platform and support what they're doing. Which is, you know, it's happening anyway but I think it would be interesting to see how the, how the Council of Europe could could could contribute to that positively. Thank you very much. I have another question here from Peter Gunning, a former Irish ambassador to the Council of Europe. This is a matter of regret that Russia's actions have led to its expulsion. This removes the convention's protection from its citizens. What can the Council of Europe and its member states do to counter trends in some states towards reducing respect for and implementation of the court's judgments. Well actually, Russian Federation was expulsed for the reasons I clearly explained and former ambassador would know that the statute is very clear on when the blatant violation of the statute happens that we, we couldn't act differently at first suspended, but it was clear that suspension was not leading to change the behavior and stopping aggression in Russia. Now, once the Russian Federation was expulsed and that was the 16th of March. The convention, Russian Federation cannot be any longer the member of the European Convention of Human Rights. However, the convention provision is that six months after it will be be a party to the ECHR so that means that until the 15th of September, whatever violation of the convention happens in that time, that the cases can be brought before the court so already we have this span of six months. And of course, we have another issue which is very important. Russian cases so called Russian cases in the court are quarter of all cases. So it's about, there has been I think at the moment where Russia was exposed to 18,000. There will be more in times to come. The court needs to decide now it will deal with these cases but it will, we will see the court will decide the manner in which they will tackle such a huge number of cases. And so there will be then issue of implement execution of these judgments. So the Committee of Ministers will continue, although Russian Federation is expulsed to monitor the execution of judgments and that is done by the Committee of Ministers four times a year in the format called Human Rights. So there will be, let's say a framework still for implementing the convention but one in one thing the ambassador is right. Russia is now the third country and the 140 million Russian citizens will after the 16th of September will no longer be protected by the convention. So the only, let's say good part of it is that we, according to international law, they are obliged to implement, execute the judgment. So when the court will deal with all those cases Russia would be, should be executing them. Now the matter is how much leverage we would have but Committee of Ministers will certainly continue to monitor the execution of all judgments that are pending. Yes, sir. Yeah, I mean look, ideally you never want any country to be expelled from an organization like this which is about trying to guarantee common standards in law across multiple jurisdictions in terms of protecting people's human rights. And it's even more regrettable that Russia, the largest state, well, physically the largest state in the world, but the largest state in the Council of Europe, having to suspend and then expel Russia. While of course it is possible to continue to take cases against them and that will happen right up until mid September. You know the medium term outlook now for the Council of Europe's ability to impact on policy development and state actors in Russia is going to be weakened because they will be outside of the fold. So, unfortunately, because of what Russia did I think the Council of Europe had no other choice. Just like other organizations that have effectively expelled Russia now really didn't have any other choice either, whether it's the Human Rights Council or, you know, or other entities that are discussing the same so you know, that is what a blatant disregard for international law and the brutal brutalizing of civilian populations. When you wage a war against your neighbor results in, you know, and, you know, I think that Russia has got to be isolated has got to be a significant cost for the continuation of this war. We are serious about trying to find a way of bringing you to an end as quickly as possible, but, but you know, the, the, the, there's no point in in pretending that the expulsion is a good thing it's not but there was no other. There was no other course of action I think available to the Council of Europe, they made a decision on the basis of the evidence in front of them and I don't think there was any other, any other choice and I think the relevance now of the Council of Europe in the context of the war in Ukraine is about some of the things we've already discussed around accountability the application of international law, how can the Council of Europe impact on the protection of citizens and civilians in in Ukraine in the context of Russian actions. And so in that way, I think the Council of Europe can can pressurize Russia but, but it's, you know, it's hugely frustrating for, you know, civil rights activists or civil society groups in Russia, who would have looked to the Council of Europe as a, as a standard setter if you like, and a, an option in terms of taking cases to court and so on that that could put Russia and policymakers they're under pressure. That's not going to be available to them now in the future. And that's, that's a, that's a huge issue but I think because of what's happening in the immediate and it's, and because of how dramatic it is. I think, I think the Council of Europe has acted appropriate appropriately and it's done all it can do in the face of, you know, of a war that let's face us not not many of us expected or predicted. Turning to your the third Irish priority culture. Minister Karen Fitzgerald, a researcher here at the IIA has the following question about refugees which I addressed to both of you. Coming up on the idea of welcoming Europe's welcome of Ukrainian refugees has provoked questions about its previous treatment of refugees, for example from Syria. Some European governments have recently sought to return Syrian refugees to Damascus. Would either of you care to comment on the perceived difference in treatment and whether the war in Ukraine has initiated a change in the international refugees approach towards refugees more generally. Not an easy question I recognize. Well, I'll give, I'll give Maria a couple of minutes there to think about the answer because I did that. I mean, look, what I would say is that you know there aren't too many positives from this war. And there are so many negatives but, but one of the positives has been the galvanizing effect of, of this war on, you know, political decision making within the European Union, and I can only speak for the European Union I can speak to the Council Europe at this stage. But certainly within the European Union, the most contentious political issue for the last six or seven years has been migration refugees and the movement of people into the European Union from North Africa from the Middle States and from other parts of the world. And anytime this issue was debated, it was deeply divisive. And anytime the Commission tried to bring forward proposals was nearly always blocked by either a group of countries, or, well, rarely one country, mainly a group of countries and there was, let's face it, an East West divide in terms of perspectives in this area. But within 48 hours, the European Union effectively agreed to literally open its doors to, to all Ukrainians and all people coming from Ukraine, Ukraine, even if they're not Ukrainian, who are fleeing conflict with no visa requirements, no quotas, no restrictions. And so far five and a half million people have made that journey. And just under 30,000 have made the journey to Ireland. So, you know, I think, I think it has shown a capacity within the European Union to, to be remarkably generous in the context of solidarity. I think in truth, many countries. Not included, but many countries, I think for many countries this war brought back really horrible memories of Russian military aggression tanks on the streets. And, and so the solidarity and the outpouring of support has really been quite remarkable, particularly in Poland, but also, you know, Slovakia, Romania, Hungary. You know, countries that would have been had very, very firm views on migration and refugees. You know, I think have have shown remark with generosity and resilience, given given the pressures that their systems are under. That posed the question, you know, why haven't we done that for others fleeing conflict, like in Syria, for example. And the honest answer to that is that when this happens on our own continent when it's happening in with our own neighbor, because Ukraine borders for EU member states. Then I think the, the solidarity has been much, much stronger. And I think maybe we need to reflect on that, because, whether you're Syrian or whether you're Ukrainian if you're fleeing from a city that's been, you know, that's been, you know, reduced to rubble. You know, the fears and the anxieties and the trauma is equally strong in both cases as it is indeed in other parts of the world. But I think, I think there is a sense that this is a European conflict that Europe needs to be to the fore in responding to whether that is offering military support to Ukraine, whether it's humanitarian support, whether it's accommodating. I think the Ukrainians don't want to be called refugees, and I can understand that they regard themselves as displaced people internationally, as opposed to the formal definition legal definition of refugee, but, but either way I think the solidarity because it is in the heart of Europe that this is happening in, and there's a European ownership I think of the solutions. And I think it's resulted in a, in a very, very different response to, for example, the, and of course, there aren't any transit countries either here, I mean with the exception I suppose of Moldova, but, but you know, even with Moldova there's extraordinary solidarity as well, in terms of helping them to cope with, with the numbers of people so, you know, I think, you know, I can understand that differentiation, although in international law. There probably isn't any distinction in terms of where you come from. But I think, I certainly hope anyway, that in the aftermath of this war. It will be easier to build consensus around how we deal with refugees from from conflicts in different parts of the world in the future. And I feel of what we've been trying to accommodate over the last three months and what's likely, unfortunately to be a significant challenge for, you know, many months to come. Yes, indeed, I'm sure we all hope that the European Union at least will draw some some lessons from that experience as you, as you suggest, Secretary General, would you like to comment. Very short because I need to play into catch but let me just continue to what Simon just said to apply on the Council of Europe side. So whether you are from Syria or Ukraine or from elsewhere. When entering the Council of Europe legal space you are protected by the European Convention of Human Rights so this is certainly the position that we have now. I also can only share the view that these terrible crisis and this the most the biggest actually refugee crisis after the Second World War in terms of numbers and the timing in which it happened. We really needed solidarity and generosity that was given by a number of Council of Europe member states, especially the neighboring but also all the other including island. So I hope also that this will actually also change a little bit and approach in all of our member states to to the issue of migration which certainly is an important issue with quite a lot of consequences. Even we have several layers on which we work with migration, I have special representative on on refugees and migrants. We have Council of Europe bank that for instance in Ukrainian case was very quick to give a grant of 5 million euros of immediate help and Ukraine is not a member of the bank. We have also recently issued a bond of 1 billion euros for seven years to assist the members of the of the bank next seven years to deal with social inclusion and other other issues. Also, we have our commission of human rights who has been one of the first to visit the countries that had been hardest hit with the refugees coming, trying to assist how to apply human rights standards, how to avoid trafficking in the beans which is often linked with with refugee crisis. We have our committee of CPT committee against torture to see that people do not end in situations which are in human and degrading. So we have a number of instruments to once the refugees and migrants are there to to overlook and monitor whether human rights and and and and convention provisions are put in place. So I really hope that all together, we will advance in the area and that to this terrible crisis that has happened will help shape better and more coordinated migration migration policies in throughout the legal space of the Council of Europe. Thank you very much. Secretary General, I know that you're under extreme time pressure so I'm going to put the last question to Minister Coveney and if you need to leave us to catch your plane. Thank you for joining us and safe travels. Thank you very much and I'm looking forward to work with Simon the team and everyone next six years. Thank you very much. Mr. May I just put one final question which is which is for audience. In the time you have left the simultaneous committee of ministers presidency of the Council of Europe and the membership of the UN Security Council for which Ireland has received of course much praise does seem like a unique opportunity. How do you believe Ireland can maximize the synergy between these these two presidencies. Gosh, well, work really hard that's the first thing. And that's what we've been trying to do on Security Council you know we haven't been quiet. It's there are many times when it would have been easier for us to just say nothing. When we have highlighted conflict and what we regard as breaches in international law that need need to be accountable to Security Council. And I hope it was the best example of that when Ireland was essentially screaming stop that there were, you know, there was extraordinary suffering and conflict happening. And it wasn't getting the attention that was merited or needed in our view. We use the Security Council to really focus on that conflict and and other countries joined us in that I'm glad to say, but there have been many other examples too. So, you know, really, I think this is this is about consistency and credibility. So, you know, Ireland is is a country that has credibility in certain areas, and they're the areas that we like to focus. If we start lecturing other countries on military interventions. You know, it's not the space where it is credible, you know, we are credible and peacekeeping. We are credible on advising Ukraine in terms of, you know, the arms that they need to to acquire and so on. So, where we are credible is on human rights on on gender on development and education and in certain geographical parts of the world, at least our parts of Africa where we've had a presence for many, many years. And of course, in terms of peace interventions, post conflict management, humanitarian assistance. These are the sort of niche areas that are really important areas where Ireland has focused our effort in the Security Council. So, trying to prevent conflicts, trying to respond to ending conflicts, and post conflict accountability when when state and non state actors have have broken international law or conventions. And that means sometimes saying difficult things, sometimes the countries that we have good relationships with. But that's what we campaigned to be on Security Council for. And we got the support of a lot of very small countries around the world that wanted Ireland to be their voice around the Security Council table where huge decisions are discussed and sometimes unfortunately not made because of the abuse of some countries of the use of the veto. So I think increasingly credible and climate and climate policy internationally. But we need to make sure that we, you know that we deliver ourselves at home, while at the same time talking to other countries about what needs to be done. And so, I think we will try and bring the same approach to the to the Council of Europe to focus on the areas where we have credibility where where I can speak, both off the record and on the record to other ministers to try and build consensus around ways forward or interventions that are relevant to the work that the Council of Europe does. We've also got I think 12 separate conferences that we're hosting here in Ireland in different areas linked to the those three themes, you know, of sort of the founding values of the of the Council of Europe. And youth participation. And, and of course this this broader area of of welcomes and migration and how the European Union manages that so, you know, I think a lot of this is work rate, to be honest. I'm really lucky that in the Department of Foreign Affairs, and to a certain extent in the Department of Defense to. And to people who are genuinely energized by this kind of work at home and, and in particular abroad. And that's why, you know, I'm really looking forward to meeting the Ministers of the Council of Europe in in Turin in a couple of weeks time and to I think, trying to ensure that Ireland is seen as as a serious chair of those discussions. And focusing on the issues that are relevant and where where we're credible but like make no mistake. The next six months is going to be all about Ukraine. And it's the war there and its consequences, both legal human military and political. You know it is an unfolding human tragedy on on a massive, massive scale. And, and we need to find ways of using all of the bodies that we happen to be sitting on, and we happen to have the privilege of having leadership positions on the moment, to the greatest extent possible to ensure that that Ireland, a small and a large part of Ireland has a has a strong and credible voice that can that can be a catalyst for change. But the final thing I'd say is that what we are good at generally in international politics is bringing people together, you know, and taking views that that involve compromise and then trying to be as persuasive as we can be to build sort of coalitions of effort around around some of the work that we're doing. And that's the approach here now as well. And I'm lucky to be working with Marie actually I know her very well from when she was Foreign Minister and Deputy Prime Minister in Croatia. So I'm looking forward to working with her and her team she was in Dublin a few days ago to plan the six months ahead. And, you know, we'll, we certainly won't be found wanting in terms of effort. Building consensus in the Council of Europe isn't easy actually because there are a lot of very different perspectives, much broader than the European Union. But I think, I think we'll certainly give it a give it a go and try and be as impactful as we can be over the six month period that we have in the chair. Yes, you have a challenging agenda and a busy one at a historic time in Europe, we wish you every success with that. And thank you so much Minister for joining us for this. It's going to be a busy summer. Thank you on the Council on the on the Council of Europe within the EU and it looks like in Northern Ireland to so you know anybody who says this that foreign affairs is a kind of a calm easy brief. I can tell you, it's, we're getting, we're getting a series of challenges that that you normally wouldn't get in four or five decades in the space now of of a short number of months but it's, it's something that I that I think we're equipped to take on and and be credible on. I'm sure Minister, given what you've achieved so far you will rise to that challenge. Thank you so much for joining us. Thank you to all who listened and watched, and thank you of course to the Secretary General for her contribution. And I now will close this session. Thank you.


So we were talking about morphisms. So the last thing we had shown, that if we look at this map, so phi i from ui to an, which is the user map that sends a point a0 to an, so here we have a i, a n and a i is not 0 because when ui sends this to a0 divided by a i until this one we leave out, that this map is a morphism, is an isomorphism. This was, we know that morphisms to an are given by n tuples of regular functions. And so we proved it that way. Now we want to give some simple, some applications. So just some facts. So first, every variety is isomorphic to a quasi projective variety. So here, I mean sometimes I had said that a quasi projective variety is just anything that we can say that we sometimes say that even if you have something which is locally closed in an, it can be viewed as locally closed in pn and so on. But here we don't, here we really mean that a quasi projective variety is a locally closed subset of some projective space. And the other statement is that every variety has an open cover by fine varieties. Recall that in a fine variety is a variety which is isomorphic to a closed subset of some an. So as I said, this statement is quite useful for theoretical purposes because it means one can prove things that are facts which are local on a variety by just proving it on fine varieties. And this is quite useful also. There is the concept of an abstract variety, which we do not consider, which is somehow motivated by this. So an abstract variety will be a variety which will be some space which in a suitable sense has an open cover by a fine varieties. So the generalization goes this way. And if one goes to the general, the modern algebraic geometry, one has a notion of schemes which are more abstract and more complicated. And you have a notion of an affine scheme, which is a generalization of an affine variety. And then a scheme is something which has an open cover by a fine scheme. So it's again a generalization of this. OK. So this is how by here things are quite simple. So let X be a variety. So if X is a locally closed sub-variety in some projective space, then by definition it's quasi-projective. So there's nothing to show. So we can assume X is locally closed sub-variety in AM. So we can put, let's say, put Y to be so phi 0 to the minus 1 of X, which is now, so phi 0 is the map from u0 to n, which is Pn is locally closed. Because we know that phi 0 is an isomorphism, in particular homomorphism. And so we therefore know that X is, and it's also locally closed sub-variety of Pn, because it's irreducible, because X is irreducible, and phi is an isomorphism. And so it's a homomorphism, and being irreducible is a topological property that is not the union of two closed subsets, unless one of them is equal to it. So this is a locally closed sub-variety, and obviously is an isomorphism. OK, so this is basically a triviality. And so the second one is not much more difficult. So we have here a local statement, so every variety has an open cover by a fine variety. So if X is quasi-projective, we have that we can write X equal to the union i equals 0n of X intersected ui. And X intersected ui is locally closed sub-variety in, or at least isomorphic, in n. So if X is a variety, it's either quasi-projective, or it's a locally closed sub-variety in n, and it has an open cover by things, which are isomorphic to locally closed sub-variety in n. So as we have here a local statement, we have to just show it is enough to check it in case X is a locally closed sub-variety in n. So we can assume, OK. So what do we have to show? We have to show it has an open cover by a fine varieties, so it's equivalent to say that every point on X has an open neighborhood, which is in the fine variety. That's the same statement, only set in a different way. So equivalent for every point in X, there exists an open neighborhood in an X, which contains p such that u is a fine. Such that u is a fine. And the fine meets isomorphic to a closed subset in some an, in this case in an. So we have assumed that X is a locally closed sub-variety in n. So we can write maybe X equal to, say, Y without Z, where Y and Z are closed sub-varieties or closed subsets in n. And now we have to take any point and find the neighborhood. So let p point in X. So then there exists a polynomial in kx1 to xn, which does not vanish at p. Maybe a non-zero polynomial, a non-constant polynomial, such that with f of p is different from 0. And always find that polynomial which doesn't vanish at a given point. So then, let me see, however, I made a mistake. I mean, I was a bit too fast. I mean, that's also true, but that's not what I want. So what I want is that there exists an f in the ideal of Z with f of p different from 0. This is because p does not lie in Z. It lies in X, which is Y without Z. So then, Z is a closed subset. So the zero set of the ideal of Z is Z. So it's clear that this is the case. So now we put take Xf, and remember what Xf is. This is the set of all points p, maybe now they're called q in X. Subset f of q is different from 0, is an open subset in X. And we see, I can actually put Yf. So you see, this is an open subset in Y. And we know that Yf is a fine, because we proved that we have a fine algebra set. We take this thing, and it contains p. And we see also that Yf is contained in X, because we take away, so f is in the ideal of Z. So the zero set, so Z is contained in the set, which we take away when we make Yf. So thus, Yf is in a fine open neighborhood. OK, so that was simple. But as I said, it is sometimes quite useful. So now we want to describe morphisms between quasi-projective varieties, again coordinate-wise, in a very similar way as we did with fine varieties. So that it's something that your coordinates are locally given by polynomials or something like that. And so let's do this, because as I said, the definition of morphism is a bit abstract and useful for theoretical purposes, but usually not for studying examples, so theorem. So let X in M, Y in N be quasi-projective varieties, and let phi from X to Y be a map. So then the following are equivalent. First, phi is a morphism. Second, phi is given locally by regular functions. So by this, I mean the following. So for all points P in X, there exists an open neighborhood U in X, which contains P. And regular functions H0 until Hn, which are regular on the whole of U with no common 0s on U, no point in projective, what we see in a moment, such that phi of Q is equal to H0 of Q until Hn of Q for all Q in U. So I can also write phi is equal to H0 Hn on U. So this is the first statement. The other, and you can see that we need that we have no common 0s, because otherwise this would not be a well-defined point in projective space. And the other statement is with polynomials. So it's locally a polynomial function, a polynomial map. So by this, I mean a similar statement. So for all points in X, there exists an open neighborhood U in X and polynomials. So maybe I call them F0, so Fn, the variables on the space we start with, which are homogeneous of the same degree. And as before, with no common 0s on U, such that the map is given in the same way. So phi of Q is equal to F0 Q until Hn of Q in U. So I can again write phi is equal to F0 F. OK, so this is rather similar to the affine case. It's also rather simple. So we will prove it by reducing it to the statement in the affine case. Now we have proven, and for this we use the result we just proved, that we have this open subset U0 is isomorphic or Ui is isomorphic to An. OK, so I mean, so if phi is a morphism, then it's also morphism as a morphism to Pn. So I can just look at that case. So if phi from x to Pn is a morphism and P a point in x, well, then it will be true. You know that Pn is covered by this open subsets Ui. So the point P will map to one of the Uis, maybe to more than one, but will map to one. So there exists an i such that P of P is in Ui. And for simplicity of notation, I assume that it maps to U0. I mean, otherwise you can manage with indices. So assume P of P is in U0. So let U be a neighborhood, open neighborhood of P such that, so P in x such that phi of U is contained in U0. For instance, I could just take the inverse image of U0 on a phi. So then obviously if we take this map phi 0, you know phi 0 was this map from U0 to n, this isomorphism, and compose it with phi from U to n is a morphism. So by what we know is, we can write it as an n tuple of regular functions, regular functions regular on the whole of U. We have this result, how morphisms to a n go. So in order to get back phi, we have to compose with the inverse of phi 0, which after was the map, which took an n tuple like this and added one to it at the beginning. So then we have phi is equal to U0 composed is phi 0 and this is therefore equal to 1 H0 H1 for every point q, we do this. And so we see these are, one is obviously a regular function on the whole of U, it's regular everywhere. And these are regular functions, so the map is given by an n plus 1 tuple of regular functions. This proves this statement. Now, 2 to 3. So assume our map is given by an n tuple of regular functions. So assume phi is equal to H0 to Hn, I mean on U. So we have an open cover, so that for every point p, there's an open neighborhood such that and so on. And the HI are regular with no common zeros. Well, we can make, we are supposed to just find for every point p and x an open neighborhood. So we can, where something is true. So we can always make our open subset smaller. You can simplify it. And by making it smaller, we can assume that these regular functions are again the quotients of homogeneous polynomials of the same degree. So the smaller can assume that HI is equal to fi divided by gi with fi and gi in k x0 to xn homogeneous of the same degree. I'm not saying that for all i, the f i's and the g i's have the same degree. It just means if I fix i, then f i and g i have the same degree. And g i non-zero g i without zeros U. We've done this many times. Now I want to prove this statement. So I just have to clear the denominators. Write that for all i, let say li equal to what? I take, I think, fi times g0 times g1 until I leave out gi until gn. So then the fi and the ni are homogeneous, all of the same degree. It's the same as the degree of the product of all the gi, because the gi has the same degree as ever. And I claim they have no common zeros on U. Let me, you can check. And we see by definition, we have that fi on U. p is equal to h0 for hn, where we have this. And now we just multiply the whole thing with the product of the gi's, which then we get l0. Because after all, multiplying by, if we do this for any p, we multiply with the value of the product of the gi on the coordinates. This is just the same constant for all of them. And it's fine. And we know that the gi don't have zeros on U. So we can multiply this without the li getting a common zero. And so this is this statement. And now finally, we have to prove that if we are given a map by polynomials in this way locally, then it actually is a morphism. So this is in 3 to 1. So to be a morphism is a local thing. If I have a map, that it's a morphism is something you have to check just locally. If you have an open cover of the source and each open subset of the cover is a morphism, it is a morphism. So let phi, assume phi restricted to U can be written as, say, h0 for hn for some polynomials as before, homogeneous of the same degree, and with no common zeros. Then the only thing we have to show is that phi restricted to U is actually a morphism. And that's so we can, by making U possibly smaller, again, we have to only find an open cover, can assume that instead of these things having no common zero, one of them actually has no zero on U. Because the set of common zeros is the section. So we can just take away the zero set to get the smaller open subset. We can assume one of the hi. So I will just assume, say, h0 has no common, has no zero. Because obviously, U is the union of the sets where one of them is non-zero. That's the statement. OK. So then we can put hi. And so for i equals 1n, we put hi equal to h large hi by h0, which is a regular function on U. Because it's a quotient of homogeneous polynomials of the same degree. Well, so in some sense, we just retrace the steps. So we therefore have that phi. In other words, phi is of the form 1h1 to hn. But that means if we take phi0 composed with phi, this is done by dividing everything by that. This is the map h1 to hn like this. And this is a morphism because it's given by polynomials. But as phi0 is an isomorphism, if something is a morphism, after I composed with an isomorphism, it was morphism to begin with because I can compose with U0. OK. I mean, to begin, in principle, we are still on U. So phi is equal to U is a morphism. And therefore, phi is a morphism. So OK. So this is some. So we have this concrete description, I mean, which is at least you can somehow write down coordinate-wise what morphisms are. It's not so super practical because you only have this open-cover substat, so you can only do it locally. I mean, but still, we have a concrete way of saying it. Just maybe a couple of examples. We want this one. So I will give a couple of examples where actually the morphism is given just by an n-tuple of polynomials. We don't need to define it locally, but there are also many where you need. So the first one is very simple. The projective transformations. Just if we have a matrix, a 0, 0 to a 0, n, a n, 0 to a n, n. So this is n times 1, n plus 1 times n plus 1 matrix with coefficients in k. Then this defines us a way, a morphism of pn to itself in the obvious way. So let a from pn to pn. I associate to a vector, say, p0 to pn. I associate this matrix applied to this vector. So I mean, strictly speaking, as a column vector like that. So I just maybe write it out like this. I just apply this matrix to the vector. And well, this would now set it on the basis of vectors. But here, I associate to the thing in projective space the corresponding vector in projective space. So this is called a projective transformation. Sometimes also projective change of coordinates. And we can see this is a morphism. Why is that? Well, we can write it down in coordinates. Because what do we see? In coordinates, it is just given by the linear polynomials. We have, for instance, the first component of the image is a 0, 0, b 0 plus a 0, 1, b 1, and so on. And so this just means that we just have a 0, 0, x 0 plus a 0, 1, x 1. That's just what it means. This is the map, which is the polynomial, which associates to b 0 to bn, a 0, 0, b 0, and so on. So this gives us the first line. And then it goes on just applying in the same way the coordinates. And this is a n 0 x 0 plus. So this thing is given by an n-tuple, an n plus 1 tuple of polynomials of degree 1. So it's a morphism. And in fact, it's an isomorphism. Because obviously, if we do it with the inverse matrix, we get the inverse map. So if we take the inverse of this morphism, is the morphism given, ah, yeah. You see, I forgot to say it. But this is not necessarily a morphism, because if we have, if this has a kernel, then this will map to 0. So all the coordinates would be 0 in the image and would not be well defined. So we have to have an invertible matrix. If the map is not invertible, this will not give us a well-defined morphism. But once it is invertible, it obviously gives us an isomorphism, because we can take the inverse matrix to define the inverse map. OK. So obviously, I mean, this is not so exciting, but it is one very often, these are somehow, if you want the most trivial morphisms which don't do anything at all, you just in some sense, really change only the coordinates on Pn. It's just if you put another system of coordinates on Pn, you would, that's more or less what it amounts to. But what does actually affect, but is actually not so easy to prove, is that all isomorphisms of Pn to itself are of this form. And at any rate, very often, you want to classify things only, I mean, to understand things only up to isomorphism, or up to isomorphism of the ambient projective space. So somehow, we often just ignore projective transformations. What? Yeah, I think it's not so easy to prove, yeah. I mean, the question is how we don't really have a reasonable criterion, how to show that something is not the morphism, OK? And so I don't think I know how, with the things that we have discussed so far, I don't know how to prove it. I mean, one needs slightly more advanced techniques, I think. I mean, I haven't tried very hard, but I think it is a bit more. So another thing that one often considers are projections. So this is, say, when x, Pn is a sub-variety, and W, Pn is another sub-variety, no, actually a projective linear, so projective subspace of Pn. So by this, I mean that you have some linear subspace of Kn plus 1, and you just look at the points in projective space which correspond to that, the image in projective space of some dimension. Then we can project from it. Do you want to do K? So we assume it does not intersect. W. Then we can project from W. So what does it? So in particular, if we have such a linear subspace, it has dimension K. So it has co-dimension N minus K. So it can be taken as a zero set of N minus K linear forms of N minus K polynomials of degree 1.  So there exist N minus K linear forms. So homogenous polynomials of degree 1, H0, so Hn minus K plus minus 1, such that W is a zero set of that. And now the projection from W is the map given by these linear forms. So the projection is, let's say, PW from X to P N minus K minus 1, which is just given as H0, Hn minus K minus 1. So for every point P, we just associate this. And this claim is a morphism. So it's given by polynomials homogenous of degree 1. So that's fine. So they should have no common zero on X. But that's also fine because the common zeros of these is your W. So the H i have no common zeros because W intersected X is N. So this is a well-defined morphism. Well, in some sense, it's not completely well-defined. I call it, here, the projection from W. But the way I've defined it, it doesn't just depend on W. It also depends on the choice of these. So PW depends on H0, Hn minus K minus 1, and not just on W. But if, say, we have another set of such forms, L0 to L N minus K minus 1, then you can see these are just two bases of the same vector space, namely of the vector space of the corresponding dual vector space. And so there will be a matrix which transforms one basis into the other. And so this means then there exists a linear projective transformation A from Pn minus K minus 1 to Pn minus K minus 1 such that it makes one into the other. So such that, say, H0 to Hn minus K minus 1 is equal to A composed with L0 and N minus K minus 1. So we see that the projection from a linear subspace is well-defined up to a projective transformation in the target. So if we somehow view them as something where nothing really happens, it's just a choice of bases, then they are well-defined. And I mean the most useful case is the projection from a point. So if P, so in particular we can, so in particular if P is a point in Pn which does not lie in x, we have pi P from x to Pn minus 1. So for instance, I mean again well-defined up to projective transformation of N minus 1. For instance, if P is the point x0, so say which one did I want? So 0, 0, 1 in Pn, then we see that we can just take the coordinate functions to define it. So we have pi P. One way to write pi P would be x0. And maybe I should say some words about projection. You can always view it like that also in this case, but I will not detail that. That you have this, you know, have x in this projective space. You somehow take a projective space of a complementary projective space which doesn't intersect this one and you make some projection to it. So by, in a linear way, so let me, so we can look at it in this case where it's a bit simpler. So for instance, if we identify Pn minus 1 with a hyperplane, just a 0 set of xn in Pn in the obvious way. So a0 to an minus 1 is identified with respect to a0 to an minus 1, 0. Then we find that if I define Pp like this, pi P of a point Pq will be equal to the intersection point of the line through the points P and q with this hyperplane. So you can somehow, so this is what you usually call projection. You know, you have a, so that the picture is really like a projection. So you have here a given point P. Here you have a point q. And here somehow is this Pn minus 1, which is actually just a 0 set of xn. And what you do is you just take the line through P and q and it will intersect this thing in one point and that's the image point. OK. So that's just one example. Later, you know, anyway, so now for the moment, that is as much as I wanted to say about this. Now we want to look at something which is in some sense simpler, but then turns out to be in the end, not so much simpler, which is products of varieties. So it's just the obvious thing. If x and y are two varieties, you want to consider the product x times y and you want this to be a variety and whatever. And why would one want to study such a thing? So one thing that one wants to use it to study is again to study morphisms because you can describe a morphism by its graph, which is a subset of the product. And this is sometimes useful, a useful way to study a morphism. You can say things about morphisms sometimes nicer by talking about something about the graph or you find it useful to prove things that way. And we will actually introduce two important notions. Using these products, which are separateness and completeness, which you know that with our risk etopology, spaces are not housed off. Therefore, they are not really compact. I mean, they were all. But we want somehow some replacement for the house of property and for the compactness. And so what we think that we will call separateness will replace the house of property. It will somehow say that morphisms from varieties behave as if they actually were morphisms between house of spaces. There are continuous maps between house of spaces, even though they are not. And one has then completeness, which replaces compactness. And so again, we'll prove some theorem that morphisms between projective varieties behave as if they were house of compact spaces and continuous maps between them. Although, again, this is not true. This is anyway what we will use it for. And this will require to look at the products in a somewhat careful way. So now I will start with the easy case, which is the product of the fine varieties. Because it's easy to see what the product of the fine variety should be. Then next time we will deal with the product of projective varieties. And the problem is that if you take x in Pn, y in Pm, then there is no projective space in which the product lives in a natural way. So you have to somehow deal with that. So let's start with products of the fine varieties. And that's actually simple. And you even had an exercise where you have solved this problem for us in some way, at least in part. So this is easy. So if x in Am and y in Am are, say, closed sub varieties, well, then as a product, we take the product. And x times y is the product of the two, which is just An plus M. And we will see that this is, again, a closed sub variety. So let's do it right properly. So let x, An, y in Am be closed sub varieties. The product of x and y is just the product. I mean, it's a bit crazy. So the set of all PQ in An times Am, which is the same as An plus M, such that P is in x and Q is in y. So that's rather obvious. And now the point is that this is also a closed sub variety. First, we can easily see it's a closed subset. So if x is a zero set of some polynomials, f1 to fn, maybe I should say as usual. So I denote so that x0, x1 to xn be the coordinates on An. And the first factor, y1 to ym, the coordinates on the second factor, Am. And then we take x1 to xn and y1 to Am as the coordinates on product. And so if x is closed, it's given we can write equal to the zero set of, say, f1 to fk for some finite number, polynomials f1 to fk. For fi are polynomials in x1 to xn and y as a zero set of some other polynomials, g1 to gl in the other variables. And then x times y is just a common zero set of these in An plus M, so x times y. This is clear from the definition because what does this mean? So the fi only have the first coordinates. So if I take a point in An plus M as a pair of pq, where p is in An and q is in Am, then the first conditions say that the point p is in x. And the second condition says that the q point q is in y. And so this is the set of all points in An plus M. The one component lies in x, the other one in y. So we then can see, so this is a closed subset. It's a bit more difficult to see. We want to see also that the product of varieties is a variety, so irreducible. So that is a bit more tricky. And we have to take some effort. We will prove it slightly more general way so that we later can use it also for projective varieties. So we prove the following lemma, which is general statement about topological spaces. So let x and y be irreducible topological spaces. Now, we want to make one assumption about the topology being somehow just a little bit reasonable. So you see, it's not quite clear what kind of topology we'll have on the product.  But now here I have a general statement. So assume we have a topology on the product is such that the following maps are continuous. So if I take, say, Jp from y to x times y, which sends any point q in y to the pair pq. So just you have the product x times y. And you just fix the point p here. And you just map every point in y to corresponding to 5 in that direction. And the other round also, so this is for all p. So it's continuous for all p in x. And also the other round, that iq from x to x times y, which sends p to pq, is also continuous for all q in y. So if I just embed this as the final direction, it should be continuous, which one would assume that most topologies one can imagine on the product will have this property. Then the product is irreducible. So if both factors are irreducible and we have this property that these embeddings are continuous, then the product will be irreducible. So let's see how one proves this. It's a bit trick. So we have an indirect proof. So we assume that it is reducible. So assume x times y is equal to S1 union S2, where the Si are closed. In other words, we then show that would then give that x is reducible. So then we have to see that, and I think we also assume that y is irreducible. And we will want to show that x is also reducible, not irreducible. So we define 1, 2, not so many, but take Ti to be, so intersection of all q in y, eq to the minus 1 of Si. So where eq was this map. So in other words, this is the set of all points p and x, such that if I take pq, this is an Si for all q in y. So if somehow this would be S1, then we see that T1 would be somehow the set of all things where all the fiber lies in it. So we know that the map Jp is continuous, now that we had assumed for all p. And y is irreducible, thus it follows that the image of Jp is also irreducible. So Jp of y, which is just p times y, is irreducible. I mean, with the topology that we have on x times y, because we know if you have the image on the continuous map of something irreducible, it's irreducible. So it follows that this thing must either be contained in S1 or in S2, because otherwise we could write it as a union of two closed subsets in the intersection. So it follows that p times y is contained in S1 or p times 1 is contained in S2. But what does it mean? It means for every point p, we have that p times y is either contained in S1. So I could also, to rewrite this, I can also it's a set of all p and x, particular distribution, such that p times y is contained in Si. This is equivalent. So this means, and this is true, for all p. For all p and x, it's either in S1 and S2. So it means that x is the union of T1 and T2. And on the other hand, I didn't say it, the Ti. So this, the IQ is continuous. Therefore, the inverse image of Si by IQ is closed. And Ti is an intersection of closed subset. So thus, it follows that Ti is closed. So it follows that x is a union of T1 and T2. And these are closed subsets. And obviously, neither T1 nor T2 is equal to x, because otherwise, the corresponding Si would be equal to x times y. Thus, it follows that x is reducible. And so we have proven our statement. So we can apply this to our, this general statement out, our concrete product of a fine, closed subset of fine space. So we assume still that, so this is the corollary. We assume that, again, x in a n, y in a m are closed sub-varieties, but actually would also work otherwise. Then it follows that x times y is also closed sub-variety. So we want to apply the lemma. And so x and y are reducible. So we have to check these two, these statements, that these obvious maps of inclusion of the factors into the product are continuous. But that's very easy. So if I take a point, so only to check, let's say, for q and y, we have that iq from x to x times y is continuous. Obviously, we also have to prove it for JP, but we can imagine that the proof will be the same. So OK, so let's do it. What is it? So I mean, note that. So we write q equal to, say, b1 to em, that's a point in y. Then iq is just a map, x1 to xn, b1 to em. It's just a map that's taking an n tuple of points in a n and sending it to this n plus m tuple, where these are just fixed numbers. And this certainly is continuous because, coordinate wise, it's given by regular functions. These are constant, they are certainly regular, and these are just polynomials, they are also regular. So it's a morphism. It's a morphism, and therefore it's continuous. So it's very simple. So this proves, and then we apply the lemma. I mean, obviously the same you would have to do, as I said, for JP, but that is the same statement. I mean, the same proof. So finally, I wanted to show you the universal property of the product, the proposition, universal property. So it's some kind of property about maps to the product which defines, somehow determines the product up to isomorphism. And it's somehow, very often, when one uses the product for something, one will actually only use this universal property. So let's see what it is. The first statement is the projections, p1, see. So we assume again, just to know that x is in an, and y in am are closed sub varieties. And we have again x1 to xn, the coordinates on an, and y1 to ym, the coordinates on am. So p1, which I can just write x1 to xn from x times y to x, isomorphism, and the same for the second projection. So it's just, and as you can see, this is obvious because I've written it down. I mean, I have given you the coordinates of the map. They are obviously polynomials. And so this is a morphism, so this is OK. And the second statement is the actual universal property. So let's add any variety. So the morphisms from z to x times y are precisely the following, the maps, which I could write fg z to x times y, which sends any point p and z to f of p g of p for f from z to x and g from z to y morphisms. So I'm saying that I have told you in some way what all the morphisms to the product are. They are given to give a morphism to the product of x and y is the same as giving a product a morphism to x and a morphism to y. And the morphisms to the product are just given by using the first morphism as a map in the first direction, the other morphism that's mapped at the other direction, OK? So I mean one can also, yeah, you can also say it differently if you want. You can say a map, so equivalently, p from z to x times y is a morphism. So this is even only if p1 composed with c and p2 composed with c are morphisms. It's the same statement, because p1 composed with c is f and p2 composed with c is g, OK? So let's prove it. So one was obvious, now two. It's also not very difficult. So in some sense. So if h from z to x times y is a morphism, so maybe then if I come and we know that p1 and p2 are morphisms, then it follows that the composition with p1 and 2 are morphisms, why do I call it h? Then we have that f, which I defined to be p1 composed with p and g, which I defined p2 composed with p are morphisms, as compositions of morphisms. And we see that p is equal to fg, because it's just a map, which sends the point to the fact that if I compose it with p1, I get f, and with p2, I get p. Just says that the map is like this, OK? So conversely, assume I have f and g morphisms. Then recall that x and y are a fine varieties, actually sub varieties of a fine space. So such morphism will be given by tuple of regular functions. So then there exists, say, f1 to fn in whole z of z. So regular function is on the whole of z. And g1 to gm in oz of z such that f is just f1 to fn, and g is just g1 to gm. Well, then it's not very difficult to see how we can write our f, g in terms of regular functions. So this is now given by an n plus m tuple of regular functions. Therefore, it's a morphism. OK, so that was maybe all I wanted to say for today. So next, so this, as you see in the case of the fine varieties, the product is a rather straightforward thing. You just take the product, and the office things work. Now we will see that for sub varieties of projective spaces, it's a bit more complicated. Because, as I said in the beginning, that if you have, say, x in pn and y in pm, you have to wonder in what projective space the product might be. And there's some trick, which is maybe a little bit of a cheat. Namely, you have an embedding. So a map from pn times pm into some much bigger projective space. And if you have x in pn and y in pm, then you define as the product the image of x times y in this bigger projective space. And then we just call that the product. And we will then see that we also have some universe property and that one can do some things with it. Anyway, we see this next time. Thank you.


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