 Welcome you to the MAP Associates Seminar, and our speaker today is Karolina Araujo from INPA, and she's speaking on symmetries in algebraic geometry and the criminology. Okay, thank you. Do you hear me? Yes. Okay, so first of all I would like to thank the ICTP math section for running this seminar and for inviting me to give this talk, and also I would like to thank all of you who came to the seminar. So the topic of my talk is symmetries in algebraic geometry, so as usual when you were studying some object in mathematics you would like to understand its symmetries, and in our case we are studying complex projective varieties, and the first more natural notion of symmetry in this case is just the notion of automorphism. So let me just remind you what this is. So we are, in general, if you have two projective varieties x and y, we have a notion of morphism between these varieties. So this is just a functional map that is locally given by polynomials. So this also leads to the notion of automorphism, and the automorphisms of a fixed complex projective variety, they form a group, which we denote by the automorphism group of x. So for example maybe the simplest example is the case when x is the projective space, then the automorphism group is just the group of PGL n plus 1c, so these are just represented by invertible matrices of size n plus 1. In general, the structure of the algebraic of the automorphism group of a projective variety actually tells us something about the variety itself. So in general, if you have a complex projective variety, then the automorphism group has a structure of a Lie group, and then we can look at the connected component of the identity, and this is actually a complex algebraic group. So that is to say it has a structure of a complex algebraic variety, and the group structure is, meaning the multiplication and the inverse are, these operations are morphisms in the category of algebraic varieties. So let us look at some examples. So let's suppose that we start with our our variety x, it's just a smooth projective curve of genus g. So we know that there is this well-known classification by in terms of the genus, so we have this trichotomy of genus 0, genus 1, and genus greater or equal than 2, corresponding to the types of the signs of the curvature, and the interesting thing is that the automorphism group of x actually reflects the geometry of the variety. So for instance, for this trichotomy, we have very different behaviors of the automorphism group. So in the case of genus 0, so this is just the automorphism group of p1, so this is pgl2, in the case of genus 1, so this is an elliptic curve, and it has a structure of, it has a structure, it has a group structure, and then with this group structure, then this connected component of the identity is just isomorphic to x itself. And for genus greater or equal than 2, then the automorphism group is spinite. Actually, we even know a bound for the order that depends linearly on the genus. So what I want you to get from this slide is that the automorphism group actually reflects this trichotomy. So we have three very different behaviors. So the first two, genus 0 and 1, even though they're both complex algebraic groups, notice that the first one is a linear group, so this is an affine group. While the second one, genus 1, this is a projective, it's a project group. It's a billion variety. So this, in general, in higher dimensions, the automorphism group will have a mixed behavior. So it will be a combination of these three types of groups. So this can be made very explicit by, you know, looking at some exact sequences, but I would not want to do this. What I would like to do next is to argue that for certain purposes in algebraic geometry, the automorphisms are too rigid and this should not be the right types of symmetry that we should look at. So let me argue by introducing the classification problem in algebraic geometry. So in dimension one, this classification problem is very easy. It was accomplished basically by Riemann. So what we want to do in this case is to classify, say, smooth projective curves classified by a model of isomorphism. So model isomorphism, we have, we first classify them by the genus and if you fix the genus, which is a discrete invariant, then for each G, you have a modular space of curves of genus G. So this is an algebraic variety and each point of this algebraic variety will correspond exactly to one curve of a smooth projective curve of genus G. So this is a very nice classification, but if we move to higher dimensions, then I will try to explain to you that this is no longer possible. So there are way too many isomorphism classes and you cannot expect to have a nice classification as in the case of curves. So let me illustrate this by explaining a very basic construction in algebraic geometry, namely the blow up of a projective variety. I will explain in the case of pitchu and then I will say what happens in general. So in the case of pitchu, so I have this, I found this very nice picture of the blow up of pitchu that I want to share with you. So this is by Statz, he made the code available in tech so you can also use it if you want. So this is what we want to do. So you start with a pitchu and you take a point in pitchu and you would like to construct another surface that looks like pitchu outside this point and you want to replace this point by a P1 corresponding to the tangent directions at that point. So this is what the picture should look like and if you want to construct it, let me just give you a formal construction of the blow up of pitchu. So we take for instance this point with projective coordinate 001 and you consider a this map, so this is this map here in blue, the projection from the point P. So this is projects from the point P to another line that does not contain P. So in terms of coordinates it can be given in this way. And notice that this this map is not defined precisely at the point P. Now if we consider the graph of this map inside, so you remove the point and you consider the graph inside pitchu cross P1 and then you take the closure then what you get is precisely what I'm calling extilda which is how I define the blow up of pitchu at the point. So you see it's easy to see that outside the point P in pitchu, this is just an isomorphism and in the inverse image of P will correspond exactly to the P1 of tangent directions at this point. So notice, so let me just say here, so this is what I just said, so if you remove this exceptional P1, so we usually call it exceptional curve, this exceptional P1 and you remove the point P then you have an isomorphism between these open subsets. So here is just the exceptional P1. And so this is going to be a very important notion that we are going to generalize. Let me just observe, this is going to be also useful later on that once you take this, you define this blow up, if you look at the second projection to P1, this will give us structure of a P1 bundle over P1 on the blow up. Okay, so this is the blow up of pitchu but in general you can, you can, I defined it in a very concrete way but you can define it in an intrinsic way and then in this way you can blow up any protective variety at any point P and even more generally you can blow up any protective variety along a proper sub variety. And so in this case you will replace each point of the sub variety Z by the normal directions of Z on X at this point. And so the important thing here is that exactly as before you define a map, so it's not defined everywhere, I mean in this case the blow up is, the map of the blow up is defined everywhere but it's the isomorphism once you remove this exceptional set and the center of the blow up. Okay, so notice that in this way we can construct infinitely many varieties that are, that look like pitchu, so they have some dense open subset that are isomorphic to pitchu but they are going to be different than pitchu. So it's easy to see that the blow up is not isomorphic to pitchu because the second batting number, whenever we blow up the second batting number increases by one. Okay, so in this way we can construct many, many surfaces or in general many varieties that look like a given variety in some open subset. And let me just point out why here we had to start with dimension two and blow up a point. So if we had just a curve, if we try to do the same thing with a curve, to blow up the point, we replace this point by the tangent directions at this point but there's only one tangent direction so we actually, you don't change the curve if you blow up a smooth point. Okay, so now it comes, so now I hope I motivated you to define this notion of birational equivalence. So two projective varieties X and Y are said to be birational equivalent if I can find two dense open subsets, U and V, and an isomorphism between them. So in general I will denote this birational equivalence or a birational map in this way and here the way we should look at this is that we, this is a map that is locally defined by portions of polynomial. So we are allowing for poles and so that's why it's not defined everywhere. Okay, so now that I have introduced, so this is the notion of birational equivalence, let me just point out that we can characterize it in a purely algebraic way. So this notion is equivalent to asking that the group, that the function field of X and Y are isomorphic over C. So the function field of a variety is just the field of functions that are, so the functions are functions to C, neuromorphic functions to C, so they're locally defined by portions of two polynomials. Okay, so now the problem, the classification problem in algebraic geometry in higher dimension becomes the following. So if you are given a projective variety X, so the first task is to find a simplest representative in its birational class. So for instance in the case when we start with pitchu and start blowing up pitchu at many points, clearly a simplest representative is the pitchu itself. So this is, for this talk I will call this just a minimal model of X. And let me just point out that in general there is not only one simplest representative, there will be some simplest representative in a given class. But we would like at least to find, to define a notion of simplest representative and be able to find this representative. And once we have accomplished this, the second step would be to construct modelized spaces of minimal models. Of course you have for that, you have to fix some discrete invariance and then hope to have a, again, this space of models to have some structure of algebraic variety. So this is roughly the classification problem in higher dimension algebraic geometry. And let me concentrate on the first, on the first step. So you are given a projective variety and you want to find a simplest representative in its birational class. So this is accomplished by the minimal model program. So the minimal model program and let me just give a quick overview, historical overview. So in dimension one, this is the classification of Riemann surfaces that I have already explained. In dimension two, so this is the case of surfaces, then this was developed, this was accomplished by the Italian school by the early 20th century. So in the case of surfaces, this situation is very simple. So all the operations that you have to do are the blow-ups that I explained. So you start with a surface and then you ask if it is the blow-up of another surface. So if it is, then you just contract that exceptional P1 and look at the new surface. So when you do this, the second batch number will always decrease and eventually you reach your minimal surface. So this was done by the Italian school in the early 20th century. In higher dimension, this situation is much harder. So in dimension three, it was accomplished by Mori in 1988. And this is actually what gave him the the fuels medal in 1990. And in higher dimensions, this was accomplished by Birkar Kachini, Haikon McKernan, or almost accomplished. And so this became a very strong techniques that we use in higher dimension algebraic geometry. And let me just say a few differences from the surface case. So the first difference is that so there are more operations than more general than blow-ups. So you have to allow for other operations, bi-rational operations among algebraic varieties if you want to get to a minimal model. And also one very important thing is that and maybe this is why it took so long to develop the theory from dimension two to dimension three is that people have to accept that we have to deal with singularities. So you cannot have a minimal model program if you do not allow for some singularities. And so part of the test is to try to identify what is the correct notion of singularity that you should allow to still have the theorems that you want and to be able to run the program. So maybe I do not want to spend so much time in that, but now once we have the minimal model program then we can now concentrate on special bi-rational classes. So let me define give you an important definition, the notion of rationality. So a projective variety is rational if it is bi-rational equivalent to a projective space. So this is the same as saying that it's algebraically that its function field is isomorphic over C to a purely transcendental extension of C. So this is the notion of rationality and so and some of the problems that are currently very much being very much studied are so which algebraic varieties are rational. So this problem has been receiving a lot of attention and we have been witnessing great developments in this in this theory, this rationality problem and then related to that if you want to prove or disprove that a certain variety is rational it's useful to have some some invariance that detect rationality. So which properties are invariant under bi-rational equivalent. So let me first concentrate on the first on the first problem just giving you a few examples and then I will start to discuss some bi-rational invariance. So this problem is open even for the simplest type of projective varieties that are hyper surfaces. So these are varieties that are defined by a single polynomial equation. So you fix D with the degree of this polynomial and end the dimension of the variety and you want to know if the generic hyper surface of degree D is rational or not. So if the if the degree is very large then it's easy to see that the the hyper surface is irrational just by computing some some cohomology groups which are invariant under a bi-rational equivalent. So the difficulty becomes for small degrees. So let me just give you a very classical example when the answer is yes the answer is rationality. So this is the example of a quadric hyper surface. So if you start with a quadric hyper surface in in projective space then you can always project from a point. So this is the stereographic projection. So if you project from this point because your hyper surface has the degree two so any line joining any line that is uh second properly second to this hyper surface will cut it in two points. So the projection if you remove this point from which you are projecting and if you remove all the lines passing through this point that are contained in the hyper surface then you get a one-to-one map into the into int's image in a in a in a hyper plate. So this gives a bi-rational map between a quadric hyper surface and and pn sorry here I should have here pn instead of pn plus one sorry for that typo. Okay so this is the case for degree two but I would like to point out that already for degree three this question is this problem is open. So if the dimension of the hyper surface is at least four then it is not known if a generic cubic hyper surface is rational or not. So it's almost unbelievable that you still cannot answer this this very simple question. So for a cubic surface is rational a cubic threefold is is not rational by the work of Clemens and Griffiths and and already dimension four we do not know. And so now let me start this discussing the second problem that I posed. So how can we detect rationality or irrationality? So which properties are invariant under birational equivalence? And now I would like to go back and talk about symmetries of projected arrival. So the first observation is that the automorphism group is not good if you want to detect a birational equivalence. So if you have an automorphism of x and you have a birational equivalence a birational map from x to y then if you compose in general you will not get an isomorphism of y. You will only get a birational self map of y. And so this is why the automorphism group is not a birational equivalence and actually this leads us to define this this new notion of symmetry that is more useful in this context which is the notion of birational self map. So if you the what I call the birational group of a variety is just going to be the group under composition of a birational self map of x. So this is an isomorphism between dense open subsets of x which do not necessarily extend to the whole space. So in particular and now introduce the so the main topic of this this talk which is the Cremona group. So the Cremona group in dimension n is just a group of birational self maps of the projected space of dimension n. So the the the Cremona group includes of course the automorphism group of pn so this pgln is contained here but it has many more elements. So in fact let me just describe you the simplest one which is which we call the standard quadratic transformation. So let me define it in in in projective coordinate so I can define it in two ways. So I can just take define start by defining it as in projective coordinates just by by taking the inverse of the of the coordinates x y and z. So this representation is good to see that actually tau is going to be an involution but if you prefer if you are bothered by this this denominators here so you can multiply everything by x y z and then you get this this quadratic representation of the of the standard Cremona transformation. So from this representation we see that actually this this map is not an automorphism so I have this picture here to try to illustrate what it does. So outside so these are these three lines in colors here are the are the coordinate lines and so the Cremona group we can see that outside these three lines it is just an isomorphism of the complement of the of the of the complement of these three lines which we call the torus. So it outside the three lines I'll try this triangle this is just an automorphism of the torus but what it does to these three lines so it it contracts each one of these lines to a coordinate point and so it is not really it's not an isomorphism. Okay so so now what so this so this is this is an element of order of order two so by a degree two so by the degree of a birational a birational self map of p n I just mean the so I can always represent it as a in coordinates by by polynomials homogeneous polynomials of the same degree without common components so this degree in this case here is true but one can actually show that you can have you have a birational self maps of p n of or in particular of p two of arbitrary degree. However which is something a classical theorem says that the whole Cremona group in dimension two is generated by the automorphism group of p two and this one standard quadratic transformation so you add this tau that I just described and then you get the whole the whole Cremona group so this may look so if you look at a representation like this you may think that this is actually very easy to describe this group you know because you have this such nice the automorphism group is a pgl three it's very well understood and then you add this one involution but then it turns out that this group becomes extremely complicated and so there are so it's it's you know it was introduced more than a hundred years ago and still people there is a lot of work being done to understand this group and only very recently we have been able to answer some some very basic question about this group so for instance only in 2010 can time let me prove that this Cremona group in dimension two is not a simple group so they produce the normal subgroups of this group and you see this is a this is very recent and also very recent only very recently we have been able to obtain a complete classification of the finite subgroups of of the Cremona group in dimension two so this is a lot of work of a lot of people so it started with the work of Bertini in the 19th century and it was only finished by recently by Dogacab and Skavsky and so this is to show that this is actually a complicated group to study and this is only for dimension two and now if you move to higher dimensions if you are interested in the Cremona group in higher dimensions then the situation is much more complicated so so Ilda Hudson proved back in 1927 that in starting in dimension three the Cremona group cannot be generated by elements of bounded degrees so this is very different from the dimension two case so the dimension two the automorphisms are linear so they have degree one and the standard quadratic has the degree two and they generate the whole group and so we cannot have such nice description in terms of generators so they cannot be generated by elements of a bounded degree in this sense and so then a natural problem is to construct since we cannot have a full description of this group we would like to construct interesting subgroups of of the bi-rational of the Cremona group in dimension n so what do I mean by interesting so first of all we would like to we would like it to have some nice group structure something that we can actually describe but also of course we would like this this group should correspond to symmetries that have some some nice geometric property for instance they should preserve some extra structure on the on the projective space so let me give you an example of such a special subgroup so this let me describe this is still in dimension two let me describe the simplistic um bi-rational transformation so if we fix here I'm writing this in a fine coordinates x and y I will write this this meromorphic volume form uh on p2 and it has simple poles sorry I forgot to write something here it has a simple poles exactly along this this triangle the three coordinate lines and we would like to describe this subgroup of um bi-rational self-maps of p2 that preserve this meromorphic volume form so this was done by Blanc in 2013 so he gave this this nice description of this this this special subgroup so so it is generated by sl2z and c star square so this is the these are automorphisms of the torus c star square and this is just another this is just one element one element of order five and so let me just explain here in blue so the sl2z acts on on p2 by this uh by monomial maps of this type so for instance here we do see the standard krimona for instance okay so now uh so this is a it's a nice group so actually Blanc also gives uh some of the relate relations in this group and we would like uh for instance one problem is to determine the the corresponding the analogous group in higher dimension so we have this natural volume meromorphic volume form in pn which poles exactly along the the coordinate hyper planes and we would like to determine the the group of bi-rational self-maps that preserve this meromorphic volume form so we do not know how to answer this although our techniques do apply in this case so um so let me uh generalize this problem a little bit more so instead of looking at the uh at that volume form that has poles exactly along the along the coordinated hyper planes let me um take any meromorphic volume form in pn so if you fix a meromorphic volume form again you can look at the bi-rational self-maps of pn that preserve this volume form now i will associate to a volume form meromorphic volume form i would like i will associate something in the uh second cohomology group of of pn so this is just taking the divisor corresponding to the this meromorphic volume form so what does that mean so the divisor of the of the form is just the divisor of zero so these are just uh correspond to the uh the hypersurfaces where this form vanishes taken with the appropriate multiplicities minus the sum of the hypersurfaces where it has poles taken with the appropriate multiplicities so this okay so this is just just just a formal linear combination of of hypersurfaces and um but of course it defines an element in the cohomology group and in this cohomology group this is always uh equivalent to this minus n plus one times a hyperplane section so and conversely if you have any hypersurface of degree n plus one in pn then you can find a unique form up to scaling a unique meromorphic form which is nowhere vanishing and it has poles exactly along this hypersurface so this is what i made by this and now the uh so given this i will just change my notation here so given the hypersurface i will denote by this this subgroup of the cremona group birational self maps volume preserved indeed to be the self of the the group of birational self maps of pn that preserve the corresponding volume form and now the problem then becomes uh if you give me a hypersurface to determine this this subgroup of the cremona group so i have been um investigating this problem in a joint problem in a joint project with alessio corte and alex masarenti and so i would like now to discuss uh some some recent results that we have obtained so by the way i we i we work this out in my last visit as an associate to ictp so i i thank ictp for that so if you um so now given the problem is given a hypersurface of degree n plus one in pn then of course determine the group of birational self maps of pn that preserve the corresponding volume form so the first result that we have is the following so if you take the the hypersurface d to be very general then you don't get anything interesting so by very general i mean that d is a smooth hypersurface and and by the left shat's principle the second cohomology group of d is induced by that of the projective space so this is that this happens for a very general hypersurface so in this case the group of birational self maps of pn preserving the the corresponding volume form is nothing but the automorphism group of pn that preserve the hypersurface d so you don't get anything interesting so you cannot so this this hypersurface has to have some sort of degeneration in order to give you something interesting on the other hand if you allow for uh to degenerate hypersurface then uh uh you do get something you may get something interesting but it's maybe very hard to describe so for instance if you concentrate all your poles in one hyperplane say at the hyperplane at infinity this will contain this group will contain uh as a subgroup the group of um automorphism of the affine space and this is known to be a very complicated group so you have for instance the Jacobian conjecture that tells us what these elements should be and and so this is uh this is this this two degenerate cases actually our techniques will not be able to to handle so the idea is to take something in between we would like to uh consider hypersurfaces which are singular but the singularities are mild and then in this case we hope to use the techniques from the minimal model program to determine this this special subgroup so by mild here for those that are familiar with the the the notions of singularities from the minimal model program this is just that yes that the pair pn and d has log canonical uh singularities okay so now let me describe uh our our first interesting uh subgroup so this is going to be in p3 and we are going to take a general partic hypersurface with one singular point um and so if you want to see the explicit equation of one such a partic hypersurface it would be like this so here uh so we take the point p to be the point with the projected coordinates one zero zero zero and then the equation of the partic is given by this form here so we see that it has a um at at the point p it will have a singularity that locally looks like the vertex of a quadric pole so this is how the singularity looks like and in this case the first thing that I want to show you is that we do get new uh non non-automorphisms in this special bi-rational so so for instance I gave you a just a very explicit one so here so here this is a rational map then if you want if you sit down you compute that it preserves the volume form associated to d let me point out that in this case when you have such a nice singularity then the the condition that it preserves the volume form in this case is is equivalent to say that it restricts to a bi-rational self map of d so this is not true in general for arbitrary d but for for this d checking that it preserves the volume form is equivalent to check that it restricts to a bi-rational self map of d and so um and so here is if you plug in the equation of d you see that it preserves these induces a bi-rational self map of d also it is an involution and you see the this uh so from the equation here the a is the equation of this this a in green here that appears here it's just the equation equation of the tangent cone at that singular point and you see that this cone here in p3 is contracted to a point under this bi-rational map so this is really not not a an automorphism it is defined um at this at a general point of the cone and it contracts the cone to a point okay so this uh so this is to show that we get something new and as I said this is a this is a this phi is a is an involution so you can this is something can also compute and and we actually uh our next in our next result we actually compute this this um by the subgroup of the bi-rational the Cremona group of p3 that uh preserves the volume form and this is going to be a semi-direct product of um this cyclic group of order two so this part here is just this uh this group that I defined in the example is generated by this phi and this g here is a this is a very interesting group so this is a form of gm over the the function field um in in two variable cfxy so what do I mean by this when I say a form of gm over this field I mean that this is an algebraic group defined over this field such that if you look if you take the algebraic closure it becomes isomorphic to gm the multiplicative group of the field and so I would like to um so if you are interested in explicit equations actually this is a sort of um a description of the group uh but here if you want to know what are the elements of the group actually we can write down all the elements of this group so in for instance in this g so these are in terms of the coefficients of the hyper surface so we can write down all the elements explicitly all the elements in this group but in a in a sense I think it's nicer to describe it in this way so I would like to uh for for the remaining of the talk to give you a rough idea of what what are the techniques that get into this uh this proof so so first of all let me just describe this this uh this basic diagram so we have our hyper surface d which with one singular point at uh one singular point at p and if we blow up so this diagram what I'm doing is I'm blowing up p3 at this point so because I have already described this similar blow up so when we do this we desingularize the hyper the hyper surface now now the the surface now becomes detailed so what I when after blowing up I get the smooth k tree surface and actually we have to understand this k tree surface um very well in order to to prove our theorem so we we understand this k tree surface the it's corresponding lattice and so on and notice that if you project from a point we have a quartic surface and it has a singular point at a point of a singular point of multiplicity two at the point p so when you project from this point actually you look if you look at this uh this morphism to p2 if you restrict now to detail that you get a two to one a double cover of p2 so this is uh this double this double cover here so now uh from this basic diagram now this is the main part of the proof of our theorem so given any birational south map of p3 that preserves the volume form associated to d we show that there is a uh commutative diagram like this so let me explain this so we blow up the point so here we look at the blow up so of course if we blow up the point we can always get um upside till the making the diagram commute so this is always true but here see the the important thing here is that this side till the is um is preserves this p1 bundle to to p2 so in other words what we have to show is that any birational south map of p3 that preserves this volume form has to preserve this car of lines through the point p so this is not true in general for any uh birational south map of p3 this is far from being true and this special types of birational south maps are called uh the genquia so we have to prove that they preserve this star of lines now once we and this is the main part of the proof uh to and I will discuss this in in the my next slide so uh but once you have this let's see what we get so now if you want to understand the group of birational south maps preserving preserving d what you have to do is to in is this is equivalent to understanding the birational south maps off the blow up x that fix the divisor the the hypers that the the hypersurface d killed them and this hypersurface maps two to one uh onto p2 so if we view our x so this x is a p1 bundle over p2 whose function field is exactly csxy so if you view x as a model of p1 over c2 so over this p1 well we have this hypersurface so this is not really two points um but after a base change actually I can think instead of having one um one detail the that double covers p2 after a base change I get exactly two hypersurface two sections and then so what we want when we look view this as a p a p1 over over the base what we have is a birational south map of p1 that either preserve that preserve this set of two points so it either switches them or fixes them uh point wise so and this is where we get this uh exact sequence so if you you see what it does to these two points whether it switches them or if it fixes them so this is the map to the cyclic group of order two actually and the and the and what we get here in the in the kernel is precisely the birational south maps of p1 that preserve these two points and the sequence splits exactly because that example that I gave you that is an involution and induces this uh this switching in the in the sheets okay so now this g is then uh bi-rational south maps of p1 that fixes these two points but birational south maps of p1 are just automorphisms of p1 and if they fix two points they're just the multiplicative so this is how we get this description of g and then from this description we can actually write down precisely what is this p1 bundle and what are the the the description of the of the elements so this is the idea of the proof so the main part of the proof is actually to get a commutative diagram that is here in uh in pink and and for that I will just uh this I will just tell you what are the tools that we use uh without too many details so the idea is that we need a program for factorizing birational south maps of pm and such program exists so this is called the sarkees of program so this was developed in the context of the minimal model program so in dimension three it was proved by corti in 1995 and in higher dimensions this was done after the mmp by heykonen mackernan in um later on in 2013 so this is the idea that if you have any uh if you're given psi a birational map between pn we would like to factorize it as simple birational maps between uh intermediate varieties so so this is what we have so this xi that appear in this the composition they are we have to um allow for other rational varieties not necessarily ppn so what what appears there in as the nodes in this graph are what we call the mori fibrous spaces so what are these so in terms of the minimal model program remember that I told you that in general we do not have only one um distinguished element in ever birational class we have a few and and the xis that can appear here are exactly um the possible outcomes of the minimal model program for a rational variety so these are uh what we call mori fibrous spaces and the psi i i will not be finding the tails but they are elementary links so they are very simple birational maps that we get that are easy very easy to describe for instance it could be just uh the blow-up of a point and and uh but of course this is uh this this is in general for any birational self map of pn and but what we have more recently uh theorem bicoarty and and calo giros is that um this program actually goes through if you're for volume preserving maps so a volume preserving birational self map of pn with a given hyper surface d um if this pair is log canonical or has mild singularities then um this this birational map can be factorized as a composition of volume preserving circuitably okay this volume preserving something that can be made more um explicit um but basically what it means is that we have um we can we can attach to each one of this x i a meromorphic volume form such that these links are uh volume preserving and um so this is actually a very this is very strong of course if you have any birational map and you have a volume form in one side we can get the corresponding volume form on the other side but in general it will pick up um singularities that get out of the of the scope of the minimal model program so here we have to do a careful analysis of um singularities and then we use this uh we use this this tools to actually prove our our main results so um so i hope i gave you uh at least an idea of what get into this uh type of problem and so i will finish here so i thank you for attending the talk and a special thanks to Santiago Arango, Charles Statz and Wikipedia for some of the nice pictures thank you thank you so much for such a nice talk and are there any questions you can either unmute yourself or write it in the chat so i can start so i i will ask if there is a similar technical or there is some information about the group of birational mofism for example of final three four or higher dimensional final yes sure i mean this this um so we so this is something that we that we have just started but of course if you take any final variety with picar number one so this is an example of a mori fiber space and we do have let me just put this again and we do have a sarkees of program for such variety so it makes sense if you want to look at other any final variety of picar number one you will have uh this the composition theorem at your disposal and then you just have to look for some interesting uh volume form and try to do the same thing so this is uh this so we i should say that it would okay it looks uh something very easy but it was actually this you have to understand in general very well the the hyper surface d the geometry of these were actually what governs what kind of link you can have so um so in so we we work this out you know for smooth for general for general hyper surface our techniques using this the composition theorem we can actually uh show that you don't get anything new so you cannot we show that the first link fails and then this was the first case that we looked at the case of partic with one double point but of course now you can look at you know what if it has more points or what if you more singular points and then in principle this can be done or you can do it for other ambient spaces as well nice thank you so much are there any other questions so claudio says that he would like to see the main theorem again ask your theorem be this one i met claudio ask sorry i for some reason i cannot see the chat now i guess he sent the he sent the question okay can i i just wanted to ask i mean up to which point this theorem now i mean i wanted to check the statements because they didn't know if it makes sense to ask whether uh so up to which point the knowing the the group of birational transformation the term means d i mean um so we have not thought about this from this uh point of view but this is certainly this is certainly uh an interesting question so for instance let me just put back the theorem a so um in general if you take something very general you don't expect you don't expect to have um anything in this group so so it will it may it will not determine d but it it probably tells you something about you but we have not we have not thought about this uh about from this this this converse question but because sorry okay let me disclose my ignorance but i i seem to remember something like typically or at least for for very high degree you expect the automorphism of p n comma d to be trivial no trivial yes correct so this is why i'm saying that it will not recover the d for instance in any d very general we'll have we'll give you the same thing here so it will not recover it will not recover the d but maybe it does tell you something about d that so for instance there there is another direction that one could look at is uh i wrote here what i mean by very general so the smooth and um and the the left sheds theorem uh holds but there has also oh oh guiso has been uh looking from a different perspective but he has been looking at k3 surfaces um quartic smooth quartic surfaces in p3 for which uh this is not true and they do have interesting very interesting automorphism group so he constructed an example for instance of uh when this automorphism group is um is a free group generated by three elements and uh and so this this is just describing this part here but i believe that in those cases this part here is much bigger and it would also be interesting to to understand what is the by the group of birational self maps of p3 preserving that quartic so there is a lot of things to do in this direction so this is something that an area that is sort of wide wide open thank you if not we again i have i have a question if you don't know somebody in the chat raised their hand i can't remember or in the thingy but i have a question if you don't mind um so in the um in the birational in the in the group you get a z mod 2z uh as factor and the the the the surface is is the surface the same thing as the sort of um the z mod 2z singularity in the ade classification yeah so this is an a1 singularity yes in general it is going to be an a1 singularity but it has nothing to do with that i think so this is just this is just corresponding this this part here is corresponding to the fact that um this if you project from the point then you have this corresponds to the fact that the birational self group the birational group of d is excellently z mod 2z because this is just given by the evolution induced by the projection from the point p okay so it doesn't have anything i wouldn't i shouldn't expect z mod nz in an a n a n minus one singularity um um no i don't think so no no no if in the k okay in the case so we are working out as a slightly more degenerate case and this is uh this is almost finished but not finished yet which is the case of an a2 singularity and you still it seems that we you still get this z mod 2z here thank you okay any more questions so thank you so much and we i would like to announce you that next week on thursday we have again the math as we say semian and he's going to talk uh camilo alias about from colombia so you're welcome to attend see next week and thank you everyone to attend our semian