On the B-Semiampleness Conjecture
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Floris, Enrica
An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli part, that contains informations on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base $Y$ is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimension.
 So, thank you very much for the invitation. I will talk about a joint work with Vlad Lazic. So, I'll start with an introduction of all the objects. In this talk, we'll have x and z projective normal and complex varieties. But more than in varieties, I'm interested in pairs. So, I'll just briefly recall you what a pair is. It's the data of a variety and a val divisor. So, it will be with rational coefficients for us today. And together, they have the property that kx plus b is q-card here. And, okay, if you're used to pairs, maybe you can notice that here, I'm not assuming that b is effective. Okay. Okay. In the objects I study are called klt or lc trivial vibrations. So, this is an notation f from x, b to z. They are the data of a pair, a vibration. So, f is a connective morphism, with connected fibers. And so, the crucial condition is this one. There exists q-card here, divisor d on z, such that kx plus b is the pullback of d. And then there are some technical conditions. We assume x, b to be respectively klt or lc. So, lc has already been defined, but let me just recall that. So, klt for Kavamata, log terminal, and lc for log canonical are regularity conditions. And they are regularity conditions that can be checked on the coefficients of b, on a suitable variational model of x. And let's say that if x is smooth, and b is simple normal crossing, then they become, so lc. So, we write b as the composition of its reducible summands. And lc becomes bi at most one for i and klt. So, we have the same inequality, but strict. Okay, and then we have another condition, which I will not tell. So, it's a... You're calling klt what is usually subklt? Yes, because my pairs are what is usually called sub pairs. So, I'm just throwing the sub away. Yes. Yeah, and then we have this other technical condition. I will not tell you what it is, but I'll tell you what it does. It makes b behave almost as an effective divisor. b is almost effective. So, four and five are technical conditions. What makes these vibrations special is condition three. So, for instance, if b is equal to zero, and if you restrict to a general fiber, you can see that a general fiber has trivial canonical. So, this is a restriction on the geometry of the vibration and of x. Okay, maybe some examples. So, the first example I want to give you is the one of relatively minimal elliptic vibrations. So, what does it mean? It means that x is a surface, a smooth surface, and c is a smooth curve, and the general fiber is an elliptic curve. So, this is an elliptic vibration and relatively minimal means, just right here. Relatively minimal means that there are no minus one curves inside the fibers. So, if f is a relatively minimal elliptic vibration, then f from x, zero to c is a KLT trivial vibration. So, this is an elliptic vibration and relatively minimal means that there are no minus one curves inside the fibers. Okay. Okay. The second example, for the second example, we're going to take x p1 cross p1 and we take D, a divisor of B degree, let's say Dk on x. Then the second projection, so from x and then I take as boundary 2 over D, D to p1, maybe D reduced. And I want also maybe D at least 2. So the second projection is NSE trivial integration. So these two examples show two very different situations. The first one where the fiber vary, maybe the fibers are not all isomorphic. And then the second situation where the fiber doesn't vary at all. So it's a product, but the interesting geometry is given by the boundaries. So the points that move in the fiber. Okay. Maybe I can convince you that this is a KLT trivial vibration. These kx plus 2d has B degree. So the canonical, it's minus 2 minus 2. And then D, it's Dk multiplied by 2 over D. So it's 0 minus 2. So it's a pullback from the base. Okay. So now we consider an S-triva vibration. And we define an object called the discriminant. So it's defined by a formula. Let me write the formula and then I'll tell you what it is. So it's a sum over the irreducible and reduced sub-varieties of what I mentioned one of z of some coefficients, 1 minus gamma p times p. And now gamma p is again defined by a formula. So we take the soup over all the non-negative real numbers such that this pair, is LC over the generic point of p. Okay. So what is this? So LC, we've already said that it's a regularity condition. Let's say for simplicity, the coefficients of Delta b plus t times the pullback of p. Let me write Delta as sum of Di Delta i. Okay. And then over the generic point of p, means that I check the this condition only on the Di such that the Delta i surjects exactly on 2p. Okay. So maybe this is not super clear yet. But I'll just tell you some properties first. So facts, Vz is a cubile divisor. So this means that the sum is finite. And the gamma p's are rational numbers. And then there is another important property proved by Ambro, saying that if Kz plus Pz is Q Cartier, and B in the boundary is effective, then the singularities of B, so R appear exactly in the singularity of the pair Z, Vz. So this manifestation of the inversion of a junction, which has been already mentioned. And then maybe I want to say two words on actually what is this? What is this? So if you don't like the formula, what does it mean? So let's say that gamma p equals 1. So gamma p equals 1 means that this pair, x plus the pullback of p, has the regularity condition locanonical. So it is regular in the category of regularity that we have chosen. And if gamma p is less than 1, then the same pair is, let's say, does not have this regularity condition, then it's more singular. So we can look at this support of Vz. So this is the union of all these, could I mention one subvarieties such that this pair is singular. So it's a sort of singular locus of F, but seen from a pair to a variety. So this discriminant detects the singularities of F. Okay. And now I can introduce another object called the moduli part. This is defined as D minus Kz minus Bz, where let me recall that D is the one from the definition which is still on the board. This guy. Excuse me. Can you find the two dimensions? No, no, no. This is a good point. Yeah, just to divisor your part. But in a second, I will tell you a result which says that maybe under some special condition, at some point it says everything. And actually, you already have the right condition on the board. Okay. So before telling you the properties of M, I would like to do a small construction. Let's say that we have here a, so F and SC trivial vibration and we consider a barational map. Then we can take the singularization of the fiber product and it comes with two natural maps. We call them F prime and tau. And then I want to define a boundary on X prime just by setting this equality to be true. So with this boundary, F prime is an SC trivial vibration. And therefore, it has a discriminant and a moduli part. And this will be, so when I take a base change, these two implicitly denote discriminant and moduli part of the vibration obtained with the base change. So what makes the moduli part special is the following result. It has been proved in different extent of generality by many people. So, ambro chevamata collar and Fugino and Gogno. So we consider an SC trivial vibration. Then we can find a barational map such that, so first, Kz plus Bz is Q Cartier or equivalently M is, Mz is F. And every time I take another base change, the moduli part, so Mz prime is F, and here Kz prime plus Bz prime. For every other barational map, the moduli part I obtain is just a pullback of Mz prime. So on Z prime, all the properties of M appear. So we say that the canonical bundle formula of F is a way of writing Kx plus B as the pullback of the sum of three divisors. So this is the discriminant which detects the singularities, the canonical bundle and the moduli part which enjoys these positivity properties. But, so maybe more than that, this moduli part is supposed to detect the properties of the fibers in moduli. So whether the fibers vary or not, if they are all, maybe all isomorphic or not. And let me give you the first example of canonical bundle formula for elliptic vibrations. So if F is an elliptic vibration, then there is a theorem by Kodaira and Gueno, maybe like this, saying that 12 times the moduli part is the pullback of a divisor of degree one on P1, where J is the J in the end. So this motivates at least the name, if all the fibers of the elliptic vibrations are isomorphic, this map is constant and therefore MC is trivial. Otherwise, it's sample. Okay. And then, maybe another important motivation, another result by Ambro. If, don't F is a KLT trivial vibration and MC is torsion, then X is a product and F is a projection. So, maybe modulo finite base change. Sorry. So, the first conjecture about the moduli part states that similar property is true all the time. So, the chain state by Prokhorov and Shukurov. And then we should be able to find a vibrational map, rho, and a natural number, such that M times MC prime, so MC prime is the moduli part of the base change, is base point three. And maybe for later, I will add, but I will denote the dependence on K with the conjecture is just for basis of dimension K. Okay. In the triviality theorem? Yes. Yes, there is, but it's KLT. It's not true for LC. So, this conjecture is wide open, is probably very hard. It has been proved in some special cases. The conjecture has been proved. So, maybe if the fibers have dimension one, so by Kudairi Ueno, and by Prokhorov and Shukurov. And for, well, if the fibers have dimension two, but are not P2, this is by Fugino, and there are some results by Shilipatsi. And if the fibers are a billion varieties, let's see again by Fugino. And now some of these results, everything apart from some of the two dimensional fiber cases, use the existence of some moduli spaces for the fibers. And there is another result, which is when mz is numerically zero. So, I just remind that mz numerically zero means that mz has zero intersection with every curve in z. And so, this is a result by Ambrov for KLT. And I proved it for LC. So, if mz is numerically zero, then it is a torsion divisor. The immediate corollary of this result is that the conjecture is true when the base has dimension one. Okay, now I've given all the important definition and some of the results. And then I can state what we proved with Vladimir Lavzic. So, I'll state the theorem in two parts. The first one is that it is enough to prove the conjecture when the moduli part is big. The second part is that the divisor is big if it's the sum of an ample and an effective divisor. And then the second part, so we assume the conjecture in dimension k minus one. And then for all sub-variate of z of k dimension one, we can prove that the restriction, um, I didn't tell you, is b semi-ample. So, what does b semi-ample mean? So, these two lines are called b semi-ample. And b stands for barational and the existence of m such that this multiple of m z prime is, is base point three is the semi-ampleness. Is it supposed to hold for every umbral model? Yeah, yeah. Yeah, no, you're right. So, we can say that z prime dominates an umbral model or one can also prove that, well, let's say, and z prime is, uh, as in the theorem. As in the theorem by Ambro, Cavamata, Collar, and Ginagobio. And then we can say that the semi-ampleness is proved not on any model, but on the one where all the important information is. Thank you. Thank you. Okay. No. Okay. I have to say something about motivation. So, the first, the first type of motivation I would like to give is more technical with respect to the result. So, the corollary, maybe of the methods more than of the result, we obtained the following. So, we assume the conjecture in dimension minus two. And I, now it's far away, but I didn't write it, but in the, in the statement, I'm assuming that the base of my L C trivial vibration has dimension K. I apologize. I hope it's clear from the contest that so T has dimension K minus one. We assume that the conjecture in for basis of dimension came and is to, and then we have F. We dimension of K of Z equal to K. And then I assume that MZ is big and the support of the discriminant, the discriminant with the augmented base locus of MZ. I'll tell you in a second what it is. Is simple normal crossing. And for every component of these, could I mention one locus. The restriction of MZ is semi ample. And this is the reminder. The plus of a big divisor is, so it is the locus where MZ is not ample. And it is defined as the intersection of all the supports of the effective divisors, such that MZ can be written as the sum of an ample plus that effective. And this guy contains the base locus of MZ. The base locus is the locus where MZ is not semi ample. So this result can be seen as an exploration of the base locus of MZ, the behavior of the moduli part along the base locus. Okay, so this is the first motivation, but maybe more importantly. So these maybe motivations to study the conjecture more than this exact result. So if we have a pair, the object canonically attached to it is kx plus b. So if we want to study the pair, we try to take this object and squeeze as much information as we can from it. So the first thing to do is look at the global sections and its multiples. And they, so if we take all the spaces of global sections, then they form a ring called the canonical ring. And now. And one of the most important questions is about finite generation of this canonical ring. And finite generation has been proved in famous paper by Birka Kashinyi, Heikana McKernan in the KLT case. And the first of all question is are the VIXB finite-generation. And some important answers have been given by Birka Kashinyi, Heikana McKernan if VIXB is KLT and K plus V is big. So then yes. And using this result, there is a theorem by Fugina and Mori. So again, if VIXB is KLT, then same conclusion holds. In this theorem, the authors use the result by Birka Kashinyi, Heikana McKernan and the canonical bundle formula. And it turns out for a lot of difficult questions, the LC case is very important. And in order to try to reproduce their strategy in the LC case, this conjecture could be really useful. So this is a reason for the importance of the B semi-ampleness conjecture. So I've got 15 minutes. So now I have a bit of time left. I can give you some ideas from the proof. So we will just, let's say, treat a baby case. I will assume that mz is big and that the dimension of z is 2. And there is a theorem by, and I also assume that z is z prime. It's a model where mz is nf and for whatever other variational map, I just obtain the pool depth every time. And there is a theorem by Nakamaya that says that the B plus, that implies that the B plus of mz is the union of all the curves which have zero intersection with mz. So I will prove, I will tell you how to prove our result for a component of the B plus in dimension 2 for mz big. And I also assume that, so just recall that we have f from x, B to z. I will also assume that the preimage of t, which I call xt is smooth. This was, of course, it's not always true, but at least it will make the proof a little easier to follow. So we write our canonical bundle formula for f. And I can add xt on the right side and on the left side. I use the junction formula. And we obtain that kxt plus the restriction of B to xt is the pullback of kt plus, plus the restriction of the divisors. So in particular, this proves that the restriction is an ac trivial vibration. So this is a canonical bundle formula. We write the canonical bundle formula and maybe the discriminant and the moduli part are different. Sorry,yes,謝謝. Have you checked the comparison? Sure, I am, you're right. I'm assuming for simplicity also that gamma p is one, gamma t, sorry, is one. Yeah, thanks. Yeah, it's not in general, of course. Okay. So we have this pullback and the canonical bundle formula. And then there is something I will not prove, but I will just let me just state as log n. So this log n is that the restricted vibration is more singular. Then the, then the vibration we start with. So of course, this sentence has to be proved, but this implies that the discriminant of the restriction of the vibration is bigger or equal than the restriction of the discriminant. Oops. But then we have the opposite inequality for the modular parts. Okay. But now, how did we choose t? T is a component of the B plus of the vibration. So this one is numerically zero. And t is a curve. So this one is an f. So these two properties with the inequality imply that n t equals m zero strictly to t. And maybe it's still on the blackboard. Yes. So up there, you can see that the conjecture has been proved if the dimension of the base is one. So this guy is semi-amplified because this one is. And so this concludes the proof of this baby case. So of course, in general, all the hypotheses I've made are not true. And so one has to, for instance, the suitable multiple of x t here, if gamma t is different than one in order to obtain everywhere as c pairs. And if this guy is not smooth, there is a sub-adjunction argument. And also this very vague sentence has to be proved with an MMP technique, which we took, we changed it a little bit, but it's very much inspired by a paper of Fugino and Gongyop. And we'll stop here. Thank you for your attention. Thank you.
 So, thank you very much for the invitation.  I will talk about a joint work with Vlad Lazic.  So, I'll start with an introduction of all the objects.  In this talk, we'll have x and z projective normal and complex varieties.  But more than in varieties, I'm interested in pairs.  So, I'll just briefly recall you what a pair is.  It's the data of a variety and a val divisor.  So, it will be with rational coefficients for us today.  And together, they have the property that kx plus b is q-card here.  And, okay, if you're used to pairs, maybe you can notice that here,  I'm not assuming that b is effective.  Okay.  Okay. In the objects I study are called klt or lc trivial vibrations.  So, this is an notation f from x, b to z.  They are the data of a pair, a vibration.  So, f is a connective morphism,  with connected fibers.  And so, the crucial condition is this one.  There exists q-card here, divisor d on z, such that kx plus b is the pullback of d.  And then there are some technical conditions.  We assume x, b to be respectively klt or lc.  So, lc has already been defined, but let me just recall that.  So, klt for Kavamata, log terminal, and lc for log canonical are regularity conditions.  And they are regularity conditions that can be checked on the coefficients of b,  on a suitable variational model of x.  And let's say that if x is smooth,  and b is simple normal crossing, then they become, so lc.  So, we write b as the composition of its reducible summands.  And lc becomes bi at most one for i and klt.  So, we have the same inequality, but strict.  Okay, and then we have another condition, which I will not tell.  So, it's a...  You're calling klt what is usually subklt?  Yes, because my pairs are what is usually called sub pairs.  So, I'm just throwing the sub away.  Yes.  Yeah, and then we have this other technical condition.  I will not tell you what it is, but I'll tell you what it does.  It makes b behave almost as an effective divisor.  b is almost effective.  So, four and five are technical conditions.  What makes these vibrations special is condition three.  So, for instance, if b is equal to zero, and if you restrict to a general fiber,  you can see that a general fiber has trivial canonical.  So, this is a restriction on the geometry of the vibration and of x.  Okay, maybe some examples.  So, the first example I want to give you is the one of relatively minimal elliptic vibrations.  So, what does it mean?  It means that x is a surface, a smooth surface,  and c is a smooth curve,  and the general fiber is an elliptic curve.  So, this is an elliptic vibration and relatively minimal means,  just right here.  Relatively minimal means that there are no minus one curves inside the fibers.  So, if f is a relatively minimal elliptic vibration, then f from x, zero to c is a KLT trivial vibration.  So, this is an elliptic vibration and relatively minimal means that there are no minus one curves inside the fibers.  Okay. Okay. The second example,  for the second example,  we're going to take x p1 cross p1 and we take D,  a divisor of B degree,  let's say Dk on x.  Then the second projection,  so from x and then I take as boundary 2 over D,  D to p1, maybe D reduced.  And I want also maybe D at least 2.  So the second projection is NSE trivial integration.  So these two examples show two very different situations.  The first one where the fiber vary,  maybe the fibers are not all isomorphic.  And then the second situation where the fiber doesn't vary at all.  So it's a product,  but the interesting geometry is given by the boundaries.  So the points that move in the fiber.  Okay. Maybe I can convince you that this is a KLT trivial vibration.  These kx plus 2d has B degree.  So the canonical, it's minus 2 minus 2.  And then D,  it's Dk multiplied by 2 over D.  So it's 0 minus 2.  So it's a pullback from the base.  Okay. So now we consider an S-triva vibration.  And we define an object called the discriminant.  So it's defined by a formula.  Let me write the formula and then I'll tell you what it is.  So it's a sum over the irreducible and reduced sub-varieties of  what I mentioned one of z of some coefficients,  1 minus gamma p times p.  And now gamma p is again defined by a formula.  So we take the soup over  all the non-negative real numbers such that this pair,  is LC over the generic point of p.  Okay. So what is this?  So LC, we've already said that it's a regularity condition.  Let's say for simplicity,  the coefficients of Delta b plus t times the pullback of p.  Let me write Delta as sum of Di Delta i.  Okay. And then over the generic point of p,  means that I check the this condition only  on the Di such that the Delta i surjects exactly on 2p.  Okay.  So maybe this is not super clear yet.  But I'll just tell you some properties first.  So facts, Vz is a cubile divisor.  So this means that the sum is finite.  And the gamma p's are rational numbers.  And then there is another important property proved by Ambro,  saying that if Kz plus Pz is Q Cartier,  and B in the boundary is effective,  then the singularities of B,  so R appear exactly in the singularity of the pair Z, Vz.  So this manifestation of the inversion of a junction,  which has been already mentioned.  And then maybe I want to say two words on actually what is this?  What is this? So if you don't like the formula, what does it mean?  So let's say that gamma p equals 1.  So gamma p equals 1 means that this pair,  x plus the pullback of p,  has the regularity condition locanonical.  So it is regular in the category of regularity that we have chosen.  And if gamma p is less than 1,  then the same pair is,  let's say, does not have this regularity condition,  then it's more singular.  So we can look at this support of Vz.  So this is the union of all these,  could I mention one subvarieties such that this pair is singular.  So it's a sort of singular locus of F,  but seen from a pair to a variety. So this discriminant detects the singularities of F.  Okay. And now I can introduce another object called the moduli part.  This is defined as D minus Kz minus Bz,  where let me recall that D is the one from the definition which is still on the board.  This guy. Excuse me. Can you find the two dimensions?  No, no, no.  This is a good point.  Yeah, just to divisor your part.  But in a second, I will tell you a result which says that maybe under some special condition,  at some point it says everything.  And actually, you already have the right condition on the board.  Okay. So before telling you the properties of M,  I would like to do a small construction.  Let's say that we have here a,  so F and SC trivial vibration and we consider a barational map.  Then we can take the singularization of the fiber product and it comes with two natural maps.  We call them F prime and tau. And then I want to define a boundary on X prime just by setting this equality to be true.  So with this boundary, F prime is an SC trivial vibration.  And therefore, it has a discriminant and a moduli part.  And this will be, so when I take a base change,  these two implicitly denote discriminant and moduli part of the vibration obtained with the base change.  So what makes the moduli part special is the following result.  It has been proved in different extent of generality by many people.  So, ambro chevamata collar and Fugino and Gogno.  So we consider an SC trivial vibration.  Then we can find a barational map such that,  so first, Kz plus Bz is Q Cartier or equivalently M is, Mz is F.  And every time I take another base change, the moduli part, so Mz prime is F,  and here Kz prime plus Bz prime.  For every other barational map, the moduli part I obtain is just a pullback of Mz prime.  So on Z prime, all the properties of M appear.  So we say that the canonical bundle formula of F is a way of writing Kx plus B as the pullback of the sum of three divisors.  So this is the discriminant which detects the singularities, the canonical bundle and the moduli part which enjoys these positivity properties.  But, so maybe more than that, this moduli part is supposed to detect the properties of the fibers in moduli.  So whether the fibers vary or not, if they are all, maybe all isomorphic or not.  And let me give you the first example of canonical bundle formula for elliptic vibrations.  So if F is an elliptic vibration, then there is a theorem by Kodaira and Gueno, maybe like this,  saying that 12 times the moduli part is the pullback of a divisor of degree one on P1, where J is the J in the end.  So this motivates at least the name, if all the fibers of the elliptic vibrations are isomorphic, this map is constant and therefore MC is trivial.  Otherwise, it's sample.  Okay.  And then, maybe another important motivation, another result by Ambro.  If, don't F is a KLT trivial vibration and MC is torsion, then X is a product and F is a projection.  So, maybe modulo finite base change.  Sorry.  So, the first conjecture about the moduli part states that similar property is true all the time.  So, the chain state by Prokhorov and Shukurov.  And then we should be able to find a vibrational map, rho, and a natural number, such that M times MC prime, so MC prime is the moduli part of the base change, is base point three.  And maybe for later, I will add, but I will denote the dependence on K with the conjecture is just for basis of dimension K.  Okay.  In the triviality theorem?  Yes.  Yes, there is, but it's KLT.  It's not true for LC.  So, this conjecture is wide open, is probably very hard.  It has been proved in some special cases.  The conjecture has been proved.  So, maybe if the fibers have dimension one, so by Kudairi Ueno, and by Prokhorov and Shukurov.  And for, well, if the fibers have dimension two, but are not P2, this is by Fugino, and there are some results by Shilipatsi.  And if the fibers are a billion varieties, let's see again by Fugino.  And now some of these results, everything apart from some of the two dimensional fiber cases, use the existence of some moduli spaces for the fibers.  And there is another result, which is when mz is numerically zero.  So, I just remind that mz numerically zero means that mz has zero intersection with every curve in z.  And so, this is a result by Ambrov for KLT.  And I proved it for LC.  So, if mz is numerically zero, then it is a torsion divisor.  The immediate corollary of this result is that the conjecture is true when the base has dimension one.  Okay, now I've given all the important definition and some of the results.  And then I can state what we proved with Vladimir Lavzic.  So, I'll state the theorem in two parts. The first one is that it is enough to prove the conjecture when the moduli part is big.  The second part is that the divisor is big if it's the sum of an ample and an effective divisor.  And then the second part, so we assume the conjecture in dimension k minus one.  And then for all sub-variate of z of k dimension one, we can prove that the restriction, um, I didn't tell you, is b semi-ample.  So, what does b semi-ample mean?  So, these two lines are called b semi-ample.  And b stands for barational and the existence of m such that this multiple of m z prime is, is base point three is the semi-ampleness.  Is it supposed to hold for every umbral model?  Yeah, yeah.  Yeah, no, you're right.  So, we can say that z prime dominates an umbral model or one can also prove that, well, let's say, and z prime is, uh, as in the theorem.  As in the theorem by Ambro, Cavamata, Collar, and Ginagobio.  And then we can say that the semi-ampleness is proved not on any model, but on the one where all the important information is.  Thank you.  Thank you.  Okay.  No.  Okay.  I have to say something about motivation.  So, the first, the first type of motivation I would like to give is more technical with respect to the result.  So,  the corollary, maybe of the methods more than of the result, we obtained the following. So, we assume the conjecture in dimension minus two.  And I, now it's far away, but I didn't write it, but in the, in the statement, I'm assuming that the base of my L C trivial vibration has dimension K.  I apologize.  I hope it's clear from the contest that so T has dimension K minus one.  We assume that the conjecture in for basis of dimension came and is to, and then we have F.  We dimension of K of Z equal to K.  And then I assume that MZ is big and the support of the discriminant, the discriminant with the augmented base locus of MZ.  I'll tell you in a second what it is.  Is simple normal crossing.  And for every component of these, could I mention one locus.  The restriction of MZ is semi ample.  And this is the reminder.  The plus of a big divisor is, so it is the locus where MZ is not ample.  And it is defined as the intersection of all the supports of the effective divisors, such that MZ can be written as the sum of an ample plus that effective.  And this guy contains the base locus of MZ.  The base locus is the locus where MZ is not semi ample.  So this result can be seen as an exploration of the base locus of MZ, the behavior of the moduli part along the base locus.  Okay, so this is the first motivation, but maybe more importantly.  So these maybe motivations to study the conjecture more than this exact result.  So if we have a pair, the object canonically attached to it is kx plus b.  So if we want to study the pair, we try to take this object and squeeze as much information as we can from it.  So the first thing to do is look at the global sections and its multiples.  And they, so if we take all the spaces of global sections, then they form a ring called the canonical ring.  And now.  And one of the most important questions is about finite generation of this canonical ring.  And finite generation has been proved in famous paper by Birka Kashinyi, Heikana McKernan in the KLT case.  And the first of all question is are the VIXB finite-generation.  And some important answers have been given by Birka Kashinyi, Heikana McKernan if VIXB is KLT and K plus V is big.  So then yes.  And using this result, there is a theorem by Fugina and Mori.  So again, if VIXB is KLT, then same conclusion holds.  In this theorem, the authors use the result by Birka Kashinyi, Heikana McKernan and the canonical bundle formula.  And it turns out for a lot of difficult questions, the LC case is very important. And in order to try to reproduce their strategy in the LC case, this conjecture could be really useful.  So this is a reason for the importance of the B semi-ampleness conjecture.  So I've got 15 minutes.  So now I have a bit of time left. I can give you some ideas from the proof.  So we will just, let's say, treat a baby case. I will assume that mz is big and that the dimension of z is 2.  And there is a theorem by, and I also assume that z is z prime. It's a model where mz is nf and for whatever other variational map, I just obtain the pool depth every time.  And there is a theorem by Nakamaya that says that the B plus, that implies that the B plus of mz is the union of all the curves which have zero intersection with mz.  So I will prove, I will tell you how to prove our result for a component of the B plus in dimension 2 for mz big.  And I also assume that, so just recall that we have f from x, B to z. I will also assume that the preimage of t, which I call xt is smooth.  This was, of course, it's not always true, but at least it will make the proof a little easier to follow.  So we write our canonical bundle formula for f.  And I can add xt on the right side and on the left side.  I use the junction formula. And we obtain that kxt plus the restriction of B to xt is the pullback of kt plus,  plus the restriction of the divisors.  So in particular, this proves that the restriction is an ac trivial vibration.  So this is a canonical bundle formula.  We write the canonical bundle formula and maybe the discriminant and the moduli part are different.  Sorry,yes,謝謝.  Have you checked the comparison?  Sure, I am, you're right.  I'm assuming for simplicity also that gamma p is one, gamma t, sorry, is one.  Yeah, thanks. Yeah, it's not in general, of course.  Okay. So we have this pullback and the canonical bundle formula.  And then there is something I will not prove,  but I will just let me just state as log n.  So this log n is that the restricted vibration is more singular.  Then the,  then the vibration we start with.  So of course,  this sentence has to be proved,  but this implies that the discriminant of  the restriction of the vibration is bigger or equal than the restriction of the discriminant.  Oops. But then we have the opposite inequality for the modular parts.  Okay. But now, how did we choose t?  T is a component of the B plus of the vibration.  So this one is numerically zero.  And t is a curve.  So this one is an f.  So these two properties with the inequality imply that n t  equals m zero strictly to t.  And maybe it's still on the blackboard.  Yes. So up there,  you can see that the conjecture has been proved if the dimension of the base is one.  So this guy is semi-amplified because this one is.  And so this concludes the proof of this baby case.  So of course, in general,  all the hypotheses I've made are not true.  And so one has to,  for instance, the suitable multiple of x t here,  if gamma t is different than one in order to obtain everywhere as c pairs.  And if this guy is not smooth,  there is a sub-adjunction argument.  And also this very vague sentence has to be proved with an MMP technique,  which we took, we changed it a little bit,  but it's very much inspired by a paper of Fugino and Gongyop.  And we'll stop here. Thank you for your attention.  Thank you.
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