 Please, the floor is yours. Okay, so thank you for the introduction and so thank you. I'm the first young speaker I would like to thank you for the school and for the possibility you give us to give a little talk over here. And today I would like to give you an introduction on five months. And, in particular, I will focus on the description of their positive and national fires. Because of the schedule I cannot take the advantage of professor figures lectures so I will have to introduce the definition of the maps and things like this and starting from the case and then moving. The three varieties are a billion varieties associated with covers, of course, and they are very interesting things they can be studied by using these covers. In particular, let's start from the definition so. Let's fix a first C in energy. And let us consider a double cover. We have a map of 5 to 1, 25, over the branch divisor B in type C, and let us assume that the degree of B is R, which is strictly greater than zero. And by Riemann orbit we know that R is necessarily E. So using Riemann orbit, so we get that the genus of B is equal to G minus one plus R divided by two. And moreover, associated with pi, we have the normal map, which is the map between the jacobian varieties. And yesterday we saw that the jacobian variety of the term is iso marked to be zero of the term so here we can say that they parametrize the degree zero divisor. So here we take some divisor of the degree two. This is the linear equivalence class and that it is sent by the normal map just to get to four more. So here we have an i pi of pi. And this is the group of momorphism, so we can consider the kernel of this group of momorphism, and this is the definition of the three-variety associated with the cover. Which is also called a block pi, which is the kernel of the normal map. And since we are considering the term five covers, his kernel is connected. The P of B of C, the three-variety, is in a medium-variety of the dimension G minus one plus i divided by two. And in case of R strictly greater than zero, then the polarization is no longer principal and this is different between the classical case, which is the one concerning ethyl covers because the three-variety is a principal polarizer with a variety while in case of R strictly greater than four, the polarization is no longer principal and I call it L. And the type of the polarization is delta, and it is one, one, one, two, two, three times. Next we can consider the mass of the map B, B, R, which goes from R, G, R. This is the start-up notation for the modeling space of isomorphism classes of covers five. And it goes to A, G minus one, plus i divided by two, equalization of type delta, which is the green marker. And the definition is the same in case of R equal to zero or R greater than zero, and it sends the class of pi to the class of its green variety together with equalization. And then in capital R, G, R, you fix the degree of the branch locus or you fix the branch locus? The degree of the branch locus. And I would like to stress that while a total three months have been studied for many, many years, starting from an analytical point of view and then moving to the algebraic point of view, which has been in some sense founded by the work of the first month of the 19th, the year before, this is R equal to zero, and this is the reason why R equal to zero is called the classical case. The 2005 case started to be investigated very, very later, only in 2012 by Marco Cipirola. And the first work, which is the one of Marco Cipirola, deals with a generic Rhele type theorem concerning three months P, G, R with R, 3, 3, 3 greater than zero. So this is the theorem G of Marco Cipirola, then completed by Marco Cinaranco for the remaining cases, and so by Naranco or Tega. And it says that if the three months of P, G, R is generically injected as soon as it is possible, which means that the dimension of R, G, R is less than the dimension of H, G minus one plus. The dimension of this model state is 3, 3 minus 3 plus R because we are moving C in NG, and this is the dimension of NG, and then we are moving also different points. And this is the dimension of the space A, G minus one plus R, G minus one plus, G plus, one half. And we have the exception of G equal to three and R equal to four. Which is, which has been showed by Barbelli, Gilbert Verra, in another work by Nagarai Ramon in 1995, that P, 3, 4 has generically decreased. And then this work, this theorem has been quite recently improved by Ikeeda for the case G equal to one and by Naranco or Tega for all the years. And I think that these works have been published here. And they proved that if R is greater or equal than six and G is greater or equal than one, then the five three months is not only this exactly. We have to stress that this is quite surprising result because if we compare it with the etal case, I hope you can hear parenthesis, it is classically it was known that PG, which is the notation for P, G, R equal to zero is genetically injected. And as it is possible, which means that we are taking G greater or equal than seven. This is the result due to treatment, and growth by kernel, and so on. But it is never injected. And this is due to the tetragonal contraction by energy. This shows that the result in case of an anti-prim up goes in the opposite direction to the one that falls in case of etal. And today, and also Naranco and Tega show that we cannot expect the result of such kind in case of R equal to two and R equal to four because they show that the prim up G2, I put here in H, and P, G4, H, which means that they consider ramified covers of verb C, lying in the hyperalytic lock in G machine. This is the restriction of the ramified prim up to the hyperalytic lock, in case of two or four ramification points as positive dimensional. Okay, so today I would like to tell you something about the joint work with Adriani and Naranco, where in some sense we complete the story because we studied the opposite direction of inequality that I wrote before. Because we studied the cases where the dimension of RGR is greater. So if I got to know that the case on C612 was on when G is equal to three and R is equal to four is greater than the dimension of A G minus one plus R divided by two. This inequality is satisfied only in six cases when R is equal to two and G is between one and four and when R is equal to four and G is between one and two. And case by case, we studied the geometry of the generic fiber, which is positive dimensional, and with adopt procedure we give a description of the generic fiber. And I think that the very interesting part of the study is the fact that when you study free mapping for low values of genus and for low values of ramification points, you have the impression to manipulate the object, to control the covariance across and so it is very difficult to study this kind of geometry. And I would like to focus on one of these cases and to try to give you the proof of the description of the generic fiber. But before I have to remark that actually the case G equals to one and R equals to four is two to half because we studied the aberrant surface with polarization of typhoon. Okay, so from now on we will focus on the case G equal to two and R equal to two. So we study B to two going from R to two to A to two. The main two are the polygonal construction. Before I mentioned the tetragonal construction, the tetragonal construction is one among the polygonal construction, which are a very useful tool in case of green theory because they allow you to start from a tower of fours of such kind. So this map is two to one and this map is two to one. And then you produce other towers, A prime, B prime and C prime, of paths of the same degrees, two to one. And also the polygonal constructions produce relations among different varieties. In our case, we, in this case, G equal to two and R equal to two, we focus on the polygonal construction and I would like to give you some details on this construction. This works in this way. So let's start from a tower of fours, two to one, two to one. This one, this map is five, this map is F. Then the polygonal construction produces another tower of fours of the same degree, two to one, two to one, this map is F prime, and this map is F prime. In this way, you see this G12 inside the symmetric product of C twice. Then you have five, twice, and here you have two, twice. And then the three images of this, you guys, one, two, two, five, one. Inside the twice is the tower of D prime. So if you start from a point B here, then considering F, it has two three images, P1 and P2, and both have two three images considering the last five. A1, A2, A1, and the two. So D prime has elements as a map of four to one, two to one, because it sends A1 plus D1, A1 plus D2, and all the other combinations to the point P in P1. And also it carries an involution, given considering it's changing the shift from A1 to A2 and from A1 to D2 and so on. And so we go to C prime, to the quotient of D prime via this involution, and then we get the other two to one. And the regular construction is useful because it allows you to understand one more thing about this tower. We know that the genus of D prime is equal to G plus R minus two in the genus of C prime is equal to R divided by two. So you know that P, DOC, and P of D prime, C prime are dual of the young variety. The regular construction is a symmetric, which means that if you start from D prime, C prime over to one with this map, then applying the regular construction again, you re-get the tower to start with. And finally, it extends to the boundary, which means that if you have a situation like this, you have the point of P here in P1, when you have two elements in the five to F is a tile over P, but Pi is unified over the two images, and you produce an admissible color. And this is the situation star, and you can also do the reverse. Okay, so our theorem is the following, the generic fiber of P22 over this generic in P2 is isomorphic with projective plane minus 15 lines. The idea is the following, we start considering between map to R22, and then we go to A2, and here we have S, and we would like to study the fiber of P22 above S. But then we know that every element B over C in the fiber has a C of genus 2, so it meets a natural P21 map of P21, which is the algorithmic evolution. And then we can apply the regular construction going to be R22. And we know that the regular construction produces a tower, the prime, C prime has genus C0, since I said that the genus of C prime is equal to R divided by 2 minus 1, so if I want, and this is if I want. We can see a tower of this type as an element in the modular space of two collections of points in P1, the first one given by six points, since we know that the branch locus of prime is given by six points, P1 plus P6. So we have to describe the S map, and we consider this map as described by choosing two points in the line, which give the ramification locus of the map of f prime. So we know that we can go to the boundary, and the boundary here is identified by allowing the two points here coming together. So we can put here a line, so considering the sort of partial ramification. Then we compose with the map of f prime, which is the forget the map, so we forget the two points. Then we compose with J, which is just the first map. So we know that we have to study the fiber of P22 above S, we know that this map is inductive. The map B is an isomorphism on the inner wall, because we know that it is inductive, and if we are straight to the image, then we can also do the reverse and the reverse is the regular construction again. So it remains only to study the fiber of if we see S as the Pythagorean on the nitrolytic curve H, then we see H here, as described by the six values first point, then we have to recover all the possible collection of two points lying in the fiber of P over H. But the point here is the crucial point is the description of these values, because we know that we have to understand to control some sense of all these copies of points, because when we, our goal is to recover the element in the fiber of P22 above S. And thus, when we have an element here, we would like to apply the regional construction and to get an element here. But the point is that we cannot admit all the possible copies of points, because we don't want that applying the inverse of the regional construction, which is the regional construction. We don't want a situation like a situation, because it would produce an element which will lie in the boundary of R22. So, to do that, we describe MAP F prime very genetically, because we see P1 inside the P2 via the random embedding of the phonics. MAP F prime, which has an implication of this described by x1 point x1 and x2, can be described as follows. MAP F prime from P1 to P1, then x1 to x1, x2 to x2, then we consider the pole. And I find this slide that considering the projection from the pole. So, here we have z1, which is centered to z2, which is this point here. So, z1 goes to z2. And a configuration like this corresponds to a harmonic crash. We have a harmonic crash here, x1, x2, z1, z2. You can combine this one. So, in our case, we don't want, we have six points of pixels in P1. And we don't want Pi and Pj going via prime to the same point in P1, because this would produce a situation like this, which does not belong to R22 via the regional construction. And so we can, if we call the x-mobile space M0621 mod. We have that, we have that M062 bar mod, which is given by plus plus P6. So, x1 plus x2, belonging to M062 bar, except x1, x2, Pi, Pj is different from one each one, for all I different from there. And we, taking into account the diagram, we have that S is identified with H, which is identified with the six point P1 plus P6. And we have to recover all the collection of points, x1 plus x2, lying in the fiber of H via Pi. But then, if we have x1, x2, Pi, Pj equal to minus one, means that we would have a situation like this. This is Pi, this is Pj, this is the pole, and this is the section of the line, of the line, which goes out from the pole, Pij. And we don't want all these configurations. And this means that we allow all the points x1 plus x2 in S2 of this method product of the line twice, and we identified as a point in the P2 dual, because we identified a line in P2, which is a point in P2 dual. So again, we have to avoid all the points x1 plus x2, which correspond to a configuration like this. So we have to avoid all the lines which pass through the pole Pij. This means that we have to avoid all the lines in the dual space, the Ij, these are 15 lines. This gives the genetic interpretation of the generic fiber in case of P2 dual. The question is similar, we put the three marks of PGR inside the communicative diagram, and we can play with the study of the generic fiber of the three marks in terms of the easier marks, and by means of other pre-generational sections, we lost the genetic representation. And I think that's it. Thank you, Irene, for the beautiful talk. Is there any question, or Mark? I have just one question. You mentioned that theorem by Naranjo as some other people, or maybe. So you said that the hyper elliptic locus is generally contracted, right, by the prem map. So you said that the prem map is the ramified prem map is always injective, so could you please just explain to me what is. Yeah, but there is a difference. Yeah, it is injective when you have a greater than six, but you contract the hyper elliptic locus in case of two and four. Oh, sorry, this is a four. I thought it was some numbers, some h or some. And another question, so you said that you define this D prime curve in symmetric product, and then you said in this big analysis action. You consider a question with some evolution, so I didn't understand what is the evolution, either to define a C prime. Yeah, okay, because the prime, you can consider the fiber above the point three here, you have the four points A1 plus B1, A2 plus B1, A2 plus B2, and meaning one, A1 plus B2. Okay, then attached to pi, you have an evolution, which sends A1 to A2, right? Yes. And then this one induces the evolution D prime from D prime to D prime, which sends A1 plus B1, it comes to A2 plus B2 to change the shape using I. And you really get a curve which is different from C? Yeah, yeah. Okay, are there any other questions? Okay, it seems not. Thank you. Okay, thank you again for this talk.