 Kosej? Sej, sej. in modulje spasje v smutričnih in pojeljičnih korv, in m0n bar je delinjman za komplatifikacije. In, kaj smo vzout, da je stratifizacija vzela vzela v m0n, kaj je bilo nr. noče, kaj je vzela vzela vzela vzela v komponent. Ne, da zelo nekaj njegi, vzelo je zelo vzelo, da je vzelo vzelo. A kaj je, da je vzelo kaj je, da je zelo vzelo, da je n-4 njegi, echo. in m0n bar minus m0n, which as we know is called an f-curve, and the f conjecture says that the morricon, so the cone of effective one-cycles in m0n, is generated by f-curves, so, da vzpečen graf in m0, n is a linear combination with positive coefficients of f curves. So, this conjecture is known to be true for n less than or equal to 7. This is due to mainly to Keel and McKernon and is open for n greater than 8. Tako, v te veselji, zelo zelo je zrednja uzena na statedne kompetifikace n-zero-n začeljeva, ko je v semoslju, zato si podljavno z punicim v n-plus-3, zato to ta je kden končil mozda, kde n-plus-3 in se tahle vsaj vse konec glasba po unknownj v 2. of PGL2, akting diagonally, and we define sigma m to be P1 to the m plus 3. Here I'm setting m equal to m plus 3. I'm taking the GT quotient of this base of PGL2 with respect to the symmetric polarization. A general point here is just the configuration of m to the 3 points on P1, and then m 0, m to the 3 sits inside, sigma m is a dense open subset. In general, you have a variational morphism from m 0, m plus 3 bar to sigma m, which is the identity on the interior, and contracting, let's say, a piece of the boundary. In a certain sense, the boundary of sigma m is less complicated than the boundary of m 0, m to 3 bar. In this sigma m, also, you have a stratification of the boundary, and what's the analog of an f curve in sigma m is something like this, so you take a subset of findices e1, e g plus 1, 1 plus 3, and you define a curve Li in sigma m, whose general points corresponds to a curve where the mark points marked by i collide, other g points collide in another point, and the two remaining points are separated. So this is a curve, this is a P1 in sigma m. Li is a smooth rational curve in sigma m, and we have n plus 3, and we choose g plus 1, such curves, g plus 1, and here is from 1 to g. Now I'm going to tell you what's g, such curves. So, now the question is, is the Morricon of dx sigma m generated by dLi? So, this is the analog of the Fulton-Godzecture for these spaces here. Of this thing here. So, if m is odd, is 1, and if m is even, is m plus 3. Now I'm going to tell you why. This thing here, so in that case, when it is a surface, is exactly m05. Now, let me take this variety here, m plus 2 is the blow-up of Pm at m plus 2 general points. So, what you can prove is that this sigma m is a smooth modification of this blow-up, meaning that you have a variational map between the two of them that is small. So, this thing does not contract any divisor, and the inverse does not contract any divisor as well. And the fact is that this variety here, xm m plus 2 is a murdering space. So, being a murdering space, so I don't want to define what a murdering space is, but this implies that the effective cone of xm plus 2m admits a wall and chamber of the composition, which is called Moritz chamber of the composition. So, you have the effective cone. Inside the effective cone, you have the movable cone of the variety, and I'm going to say what the movable cone is. And you have the composition in convex chambers, something like this, in such a way that this is the movable cone. So, basically movable cone means that an effective divisor, which is inside the movable cone, so it's movable, if base locus is in codimension at least two, so there is no divisor inside the base locus of the inner system associated with divisor. And this, the composition here of the movable cone, tells you what are the small modifications of your original variety. So, you may think of a chamber, the central chamber here, for instance, as the nef cone of your original variety, which is, in this case, is the blow-up, and any other maximal chamber here corresponds to a nef cone of another variety, which is a zomorphic in codimension two to your variety. So, in particular, this variety here, as I said, is a zomorphic in codimension two to our variety, so it must correspond to some of this chamber. So, with me here, we prove that this thing is, that sigma m is fun. So, the anti-canonical class of this variety here is ample. So, in order to understand what's the corresponding chamber, it's enough to locate the anti-canonical divisor of this variety here in this composition. He will lie maybe inside the chamber, maybe on a wall or many walls, and if it lies inside the maximal chamber, that chamber is going to be the nef cone of sigma m. So, here two things happen. If m is 2g-1, so is odd, this sigma m is singular, and, in particular, it's not cofactorial, and minus kx m plus 3, m plus 2 lies in the intersection of many walls. So, you have to think that, like, the anti-canonical is, for instance, here. So, your variety does not admit a smooth funnel model, and, in this case, the Picard group of this thing is z, even though the divisor class group is, this is rank m plus 3, but the fact is that you have many vile devices that are not karteer because of these singularities. But, if m is equal to 2g, so is even, then sigma m is smooth, and minus k sigma m lies in the maximal chamber. So, it's in the same maximal chamber, and this maximal chamber, as I said, is going to be its nef cone. So, this is what g is. So, now, if you want to answer to this question, so, we know that we have this many f-curves in this space. So, of course, now this question makes sense just when m is even, because when m is of, the Picard group is z, and there is nothing to say, OK, all the cones are degenerated inside the Picard group. So, we have 2g plus 3, 2g plus 1 lies. And, indeed, the more it's under the composition, so, this wall and some of the composition that I pictured here is known for this particular modern space, for the blow-up of Pm in m plus 2 points. So, this has been studied by many people, for instance, myself and Carolina Raugio, and Mukai and Castrovett and Pevelev. So, the more it's under the composition of the effective cone of xm m plus 3 is known, and the nef cone of sigma m is exactly that number, extremer faces. So, duoli, this says that the more it's under the composition has that number of extremer faces, because then the more it's under the composition, is the dual of the nef cone, OK? So, now the only thing that is left to prove is that these classes here are indeed extremer. And in order to do that, you have to consider a very natural morphism from sigma 2g, so the even version to the odd version, OK? This is just the forgetful morphism, one of the forgetful morphisms, forgetting a mark point. And as I said, this variety here is singular. There's, funnately, many singular points. And this morphism here is a vibration of sigma 2g. And the general fiber of this vibration is just a P1. But on the singular points, if you look at the preimage of the singular point here, you will see that the preimage is the union of two PGs. So this is something that looks like a PG union, another PG. This Li here can be realized as the class of a line inside one of these PGs. And the other one is going to be another, let's call it LJ, OK? So, what's a curve contracted by this morphism? So a curve contracted by this morphism is either a general fiber, OK? Or a curve, reducible curve, that splits as the union of two reducible components, one in this PG and the other one in the other PG, OK? But since they are in this family, you can deform the general fiber to the union of two lines, one of them in this PG, the other one in the other PG. So I'm saying that a general fiber is numerically equivalent to d times the line inside this PG and k times the lines in the other PG, OK? So I'm saying that the relative morricon, the relative morricon by i is generated by these Li's and these classes are extremely this relative morricon. But now we know that this is a general thing, the relative morricon of a morphism is extremely in the absolute morricon of the variety, OK? So, particular, these Li's generate extremely in the morricon of this variety here. So you had, you knew that this morricon had this number of extremaries by the morricon versus the composition of the blow-up of PM in m plus two points. And this argument here proves that the Li's, which are exactly the number you want to have, generate the exact number of extremaries so that the morricon, in the end, is going to be generated by these f-curves.