A Harnack's 'Über die Vieltheiligkeit der ebenen algebraischen Kurven' (part 2)

0:00 It's an example. Then we will continue to describe the content of the paper. We have seen the first part, the restriction part, on the number of circuits of the algebraic curve. Then there is a construction part, the second part, which is to show that the palm of the is sharp. So we shall see the construction in the non-singular case and seeing a few words on the construction that does in the singular case. And then we shall come to the concluding remarks of the paper, which I try to comment on there because they are rather obscure to us. Maybe we shall need the help of German speaking people to explain the text. So, first, before an example given by Hervé, I recall that Arnach establishes that for a curve of genus P, the number of circuits cannot exceed P plus 1. And the method which he used relies entirely on two main facts. One is Bézouté-Orem. He does this by constructing an auxiliary curve of degree n-2, if he starts with a curve of degree n, and counting the intersection points that he gets. And to obtain the contradiction, he relies on properties of circuits, the fact that they are divided into odd and even circuits, and the intersection property of odd and even circuits, the fact that intersection of odd circuits with node 1 has a node number of intersection points, and intersection of even circuits with any other circuit has an even number of intersections, and the fact also that if you have a curve of degree n, then

2:30 the number of odd circuits of the curve of degree n is congruent to n, the same parity The non-singular case means that it has exactly one odd circuit when the degree is odd and all circuits are even when the degree is even. But it can be a more complicated situation in the singular case and so we begin to recall the arguments on an example. It's a singular case, yes. So we are going to just to see an example of degree any qual form, but this time a singular one. So just imagine you have two odd circuits, so they intersect in one point, at least one. And we have, this is a reasonable, so with one singular point, the genus will be two. yes this is perfectly so the drawing until there is reasonable but imagine so assume you have an extra circuit so the parity of n being the parity of odd circuit you cannot imagine to have another So we have another oval, and so we are going to construct a curve of degree n-2 equal to which will intersect this curve more than eight times. So we ask...

5:00 Three circuits. Three circuits, so yes. And most three circuits. So suppose that there are four circuits. And find the contradiction. So we already did it in the non-singular case, but in the singular case you just modify a little bit the argument. So you ask the curve, so you can choose five points to define a conic. So you choose this point, one point on each of them, and two points on one of these two over, so for example this one. So and you ask for, you find an ellipse or whatever, which pass through this point. So, for example... So, okay. So, but... Like that, okay. I just draw the... Okay, so for example... Oh, what a nice thing. So, but you don't have to know where it passes exactly, but when you're sure that on this Over here you have one extra point. You have to have an even number of points. And here you have also one extra point. And these branches are cutted by an even circuit. Yeah, because the degree of the intercepting curve is n minus 2, that's even. No problem, even if it is reducible, it always intercepts a circuit in an even number of points.

7:30 And so on each this odd circuit we have an extra point, and this one comes with multiplicity 2 times, so that makes 10 points of intersection, which is obviously bigger than 4 by 2. And if you have a higher order of singularities, you ask for multiple points when you cover degree n minus 2. So if you have a singularity of order 3, singularity of order 2 for you. So that's the proof that you cannot not have more than P plus 1 circuit. So this argument finishes the argument, which is explained in general case, the distinguishing between the variety of the degree of the singular curve. It occupies the end of section 1 of the paper. So after that begins section 2 on the existence of curve with the maximal number of real circuits. So, as you can see, there are two figures in the paper, the ones that you have in the French version, the two figures that you have are the original one in the original paper. I don't know whether it was common to have figures in papers, in articles, at this time. But for instance, in the later paper by Hilbert, also on real circuits of Curses,

10:00 the dark construction within the same spirit, that is very much inspired by the construction of Arnak, but in Hilbert's paper there is no figure. So first, the first construction is on, again, in the non-singular case, construct what Arnax called a general curve, non-singular, without singularity, with a P plus 1, so that's N minus 1 times N minus 2 over 2 plus 1 real circuits. So you can see perhaps just a comment about the notation. I think that it was a common notation in the 19th century for the curves. You can see some exponents. For instance, there is a formula, page 104 in the German original and which is in the French translation page 5. There is a with x in index and 2 as exponent. That means that it is a square. It means simply that it is an equation of degree 2. So the exponents are not powers. They are therefore to indicate the degree of the equation. And X is for the variables of the equation. Okay, so we can perhaps explain the construction in the singular case. So we are going to perturb this, so we start with, okay, for the degree 2 it's easy to

12:30 So in genus 0, you find, you take a circle, and it has. So we make the first two steps of the induction. So for n equal to, so the genus will be 0, you find a curve with one circle, for example, any singular . and glue, and we cut it by a line which cuts through two different points, and we are trying to perturb this, so this is the equation of this union of the line and the conic, and And we are going to make a perturbation to obtain a drawing which will look like that, two circuits. So the degree will be 3 and the genus will be 1. And this circuit will be cut by a line to the same line three times. So how to do that? So, we are going to make a perturbation of this by a term of degree 3, which will be the product of 3... Non, elles sont notées comment ? QX1, QX2, QX3 ? Ah, des parenthèses, parce que ce n'est pas le dégât. So, by a product of two lines, three lines, so you choose this line to be, you choose this line to be there, so this is Q1, Q2, Q3, and you choose, And you take this, so this is a family of curves of the degree 3, and you choose an infinitely small value of lambda, so it really says an infinitely small, it says in a unendlich

15:00 So the curve will be infinitely near to the project of the line by the circle. So it says, ,, which is an indisputable neighborhood. Immediate neighborhood. So the perturbation, so the resection curve will be cut. So it's easy to see here. But it will cut the line VX in these three points there. So it will be done something like that and here it will be very near to the circle and take a line and this part will make something like that. So here my drawing was this. You cannot talk to us. Why... Why... ...the... ...go outside the... ...the... ...improvement? Well because... It stays on always on the same side of the union, of the union of the, of the conic and the, and the, it may only cross when the, the second term here vanishes, may only cross when it, when it intercepts the line. Okay. The only points where the product AX by VX can be crossed by the new, so the red curve, are these three points. When the second part of the equation finishes,

17:30 One of the equations of the line finishes. Otherwise it stays always on the same side of the union of the conic and the line. In the drawing, the degree seems to be formed. Why formed? There is a line that cuts the red in 4 points. And then it's always up there. No, but that's an artifact of the drawing. You can see on the questions that you have. I don't know what that is, but I think it's going to go to the top. Yeah, that's right. It's not possible. It's not possible. Yes, it's perfect. No problem. But this has to be very near. The infinite point for me. What is the control? It's not here. If you don't want to go back to the infinite point of view, because if you don't want to go back to the right, you can go back to the right one, two, and then one. So when you use of the drawing there, I'm trying to draw a degree. If you take this, if you take the... Then it will cut four times. So if you look at the figure 2, it's clear that it's not going to be... Yeah, it may. Well, yes. Ok, it goes to a point which is rather near to the infinity, the direction infinity is near to close to a reason.

20:00 So it's not asymptote to the right, but it's something that's very projected to the right, but not enough for... It's a direction asymptotique, to my opinion. That's a little direction which is close to... Yeah, it's a little bit more. But this is very... You see that the perturbation here is very small when you are in this region. This is the product of the equation of the three lines. So here, it is very close, and here, it can be more loose approximation. Yeah, that's how we're doing. Yes, yes, that's true. Okay. So we are in this situation. And... Je la mets où ? I'm going to go to the next step. I don't have enough space to put it on the right side. I'm going to put it on the right side. I'm going to put it on the right side. So this is a curve of degree 3, with two circuits, and one of the circuits, actually the odd one is cut in three parts by the line. And you take four lambs this time, so Q1, Q2, Q3, Q4, and again you are looking at these, for instance, completely small lambdas.

22:30 You have the point of intersection with the line at these four points, again, well, again, Again, these are the original points for detection with these lines and the horizontal lines. And you will obtain something like that. So this is in fact an event here, which is coming there. So if you choose carefully the sign of lambda, you will obtain two or half there. Is there something wrong? On the black, on the exterior. On the interior. The same thing, the same thing. So we are facing the 4 and the 3, right? So that's the curves of the fourth degree with four circuits and one of the circuits is cut by... same line which is still here from the beginning by in four points and which is the hypothesis on which you and so he continues a step further and he says that afterwards that

25:00 there are only two things which matter for this projection I am reading Arnaque in English, I think it's on page 6 in the French version, and on page I've lost the page 4 of the German version, but so I'll just tell you what it is. It may be at the end of page 6. So there are only two things which matter for this progression. Firstly, there must be among the p plus 1 circuits of the curve of degree n, at least one, which is cut by a line in v sub x in n real points. And then, secondly, the n plus 1 lines, Qx, with superscript parentheses, must be chosen in such a way that the node cuts the fixed line Vx in the interior of the n minus 1 segments, n minus 1 finite segments, which are delimited by the intersection points of the curve of degree n. Here you see the foreline cut the baseline, which is always here, outside of these two segments, the finite segments, which are delimited by the intersection point. And That's always the principle of the choice that you have to make. When you want to construct the curve of degree five, you have to cut with five lines. But the five lines that you will choose will cut the best line here outside of the three segments here,

27:30 which are finite segments limited by the intersection point. Then you can continue with the same principle of transformation. going up in the degree to construct for every degree n a curve of, so non-singular curve of genus, so n minus 1, p equal n minus 1, n minus 2, over 2 plus the genus p, having P plus 1 circuits. And by the construction, you can always assume that one of the circuits is cut by the line which is still there in the end point. OK? So just a few words on the... So that's, of course, construction in the non-singular You don't get, for non-singular curves in the plane, you don't get every genus, since the genus that you get are of the form n minus 1 times n minus 2 over 2. So, for instance, you cannot get genus 2 in this way. That's the formula for the genusophones, not the plate curve. So you have to... So you have to show that you can build a singular curve of degree n with up to n-3 double points. And if you show that you can do n, so n with p plus one, so the genius will be n minus

30:00 So this will be the genus of the card. And this can be done, so the idea is the same. So you start, so you make the induction, you assume you have that for n and you have such n so for small values of n you can do it by one hand and you if you have that you show that you can find a power of degree n plus one with one more singularity and one more frequency What's the number of, well, with a good number of it? So you see, the point is that you have to, well, Max wants to fill the gap between the thickness of the non-segular curves, so recall that each, say, double point decreases the genus by one, construct curves with a good number of double points, and again the construction is by induction on the degree of the curve, so we suppose that for curve N minus 1, yes, like this, for curve of degree N minus 1, he has constructed all curve with

32:30 double points and with a good number of circuits, with maximum number of circuits, and he proceeds to the curve of degree M, and as you, actually that's N to N plus 1 what he does, but nonetheless. And you can see on page 7 in the French translation, on page 197 in the German original, you have the equation of the curve of the N plus 1. It's of the form A to the N times V. So that's again a union of the curve which has been constructed of WN with a line. And this is again perturbed. And this time it is perturbed by the union of another curve of WN he constructs, taking into account the double points of the curve, of the preceding one, and a line in double units. So here the perturbation process is more complicated than in the singular, non-singular case. But that's the same idea, small perturbation of the preceding curve of degree n, of the But this time the perturbation is more complicated, and the argument is more delicate, perhaps not worth entering into more details, because it could be rather long to be very delicate. Difficult to draw, to be growing, is that clear? It's difficult to make drawing, but that's a clever perturbation, clever construction. And so we end this part by saying that with this consideration, which can be done in, which can be modified in several ways, so that's not this process of construction, it's not unique, you can think of, but that's what you get, this.

35:00 So, following a remarkable property of algebraic curve is now completely established that for every genus P, there are curves with maximal number P plus 1 of the circuits. So that's the end of the proof. And the paper close with two paragraphs of remarks. And, so, I wanted to comment on this paragraph, although I am far from understanding what ANAC means. The first paragraph says that such curves, the curves with the maximum number of real circuits can, may present an important advantage for the research on algebraic integrals and on the functions which are deduced from these integrals by inversion of the integral. that functional correspondence which appear more completely in the real domain for such curves and for a curve with fewer real circuits, and for instance, say for a non-singular curve, I think here is my non-singular situation. Those are the two key periods of an integral on the curve. They have the, for a curve with many real circuits, they have the geometric meaning. Maybe I can make some drawing at this place. I don't know what it says. So you have a curve of genus.

37:30 So I'm drawing the complex part here, that's a curve, that's a torus with three holes, that's a curve of genus three. and so the 2p here are six barriers corresponding to the six closed paths on these stories which generate cycles which generate the homology and here if you have a curve with maximum number of real circuits, that is in this case 4, since the genus of the curve is 3, you can try to make an idea, you can draw the real circuits like this, the 4 real circuits, and imagine that the complex conjugation acting on the curve is simply the symmetry with respect with respect to this plane, this plane is cutting the torus in this four-wheel circuit. These are the fixed points on the concrete configuration, so that's a good way to have an idea of what is going on. And what Arnax says that, so the periods of integrals correspond to the 2P cycles to be closed paths of the torus, which generates the homology. And the period is, you obtain by integrating a differential form along some closed paths. This gives you the period of the integral. And it says that the two P periods can be obtained by taking on the one hand P real cycles, here, one, two, three, and other P which correspond to going from the remaining real cycle to every one of the P cycles that you have here. This is one cycle. That's four, five, six.

40:00 So one, two, three, four, five, six. These six cycles are generating in modern terms of the homology of the torus, and you can compute the values by integrating over these six cycles, and it says that the fact that this is strongly related with, here you have actually three real cycles, and the remaining three, which are in relation with the real cycles, says that it should give interesting information. And he adds also that, to be more precise, one should examine, one should further investigate the relationship between the curve of the degree n and the adjoint curves of the degree n minus 3 and also to determine the system of 2PE plus 2 times n minus 2 tangents from the points to the curve of the degree n. This I don't see at all what is the relationship with this problem of periods and real cycles. I don't understand this remark. And concerning the indication about pipes would be useful for the study of integrals of the curve to study of carriers to use these curves with many real secrets, To the best of my knowledge, I don't know of any state in this direction. I'm not aware of any results, any paper related to this suggestion of Arnak. But maybe, of course, I have very limited knowledge in this area, so maybe there is some work in this direction, but nothing that I know. So that's, it says to be checked, but it seems to be some dead end, actually. But I think it's interesting to notice because it shows that this problem of Abenian integrals

42:30 and so on, and periods of integrals, is very important, I think, for the algebraic geometry at this time. So it's important to have some remarks relating to this problem in the paper. Maybe it can be useful for this study. It was the first concluding paragraph. So the other concluding paragraph also I must say that I have difficulties with Germany. I don't know. I find it rather difficult to read. Well, I'm not very fine. In general, and especially the concluding paragraphs, I don't know how it looks for or to a German native. I can read it, but it's another question to understand. But even the linguistic construction is rather intricate. For instance, it's much easier for me at least to read paper by Artin and Schreier This is more coincidental, just part of it is maybe that this is older children. Yeah. But even in the, if you take papers by Klein, I think it's easier. not read very much, but look at some, it's easier for me to read clients than to read out, from the linguistic point of view. So here the phrases are rather long, and to understand the constriction of the sentence is not quite easy. So the last three microscopes are very obscure points, very obscure to me at least, but what I catch is that this preceding consideration has also to be extended in the direction of research about how the different circuits are, the relative disposition of the different circuits at one point.

45:00 So I see here some point in the direction of what Hilbert first part of Hilbert's problem about the disposition of ovals of plane curves, the relative disposition of ovals. So if you are just considering the real circuits by themselves, you know that there are circles, you know the maximal number so sometimes everything is known but the point is that if you consider what happens for the relative disposition in the plane in the projective plane of course when you have two for example two ovals they can be apart like this or one can be inside the other and what are the possible predisposition of Orvalds in the plane is the first part of Hilbert's 16th problem. So that's some, with this concluding paragraph, it goes in this direction. And also there are some considerations about the influence that this position can have on how the number of circuits diminishes, how you can erase circuits in some sense, but that's not quite clear what it means here. But what is rather clear is that he considers this problem of writing this position a formal, which is not at all considered actually in his paper. Okay, so that's the two. End of what was already... I just wanted to say that I was surprised to find an infinitesimal in this paper because in the official credo of the Church of Mathematics, we are in the post-Cauchy era, so you should

47:30 not find infinitesimal there, it's either too late or too soon to refer to infinitesimal. a modern reader you translate it easily to something small enough but that's not what's written so it's very common in this time it's really very common idea of post-cauchy doing mathematics just wrong okay it's not an isolated case okay really common Yes, yes. I don't know. Everyone will be surprised. One comment I will make about the content is that, well, before reading this paper, I knew, well, I knew the main argument in the non-singular case, the argument using Bezoot theorem to show that there are no more than B plus 1 circuits, and the Arnax construction in the non-singular case with the drawings that Harvey made. Well, this you can find in survey papers on the topology of Remerge-Brankers of, well, maybe not very recent, but not, let's say, of the end of the 20th century. This you can find, this is still there, some classical facts, first facts in the study of Euronate-Brite curves. Also the structure of the paper is also very present, for instance, when you look at papers, especially survey papers, geometers of the Russian school, which are studied the first part of the problem and extension. It makes a distinction between restrictions and constructions.

50:00 Restrictions correspond to what is done in the first part, it's to obtain some bounds on topological invariants, which prohibit for curves of a given degree, prohibit some possibilities bound, for instance here, bound of the number of ovals which are prohibition. You cannot have a curve of degree n with more than n-1, n-2, over 2 plus 1 circuits. bound, the first part of the paper, and there's the restriction part, and the other part is the construction part, to make construction to show that, if possible, that all possibilities which are not prohibited by the restriction can actually be obtained, obtained. And hopefully you can fill the gap between the restriction on one side and the construction on the other side to have the list of all possible situations and that you have no case left. That's what is done in the first example of this strategy in this Arnax paper, so that's kind of a model for the work which will follow the problem that you have the restriction construction and if possible, if nothing between the restriction and the construction. In the case of number of circuits, it's what Darmann does. So that's a kind of model for future, for strategy for future works. And actually, from the point of view of restrictions, there has been a lot of, well, other topological which I consider and for which is a restriction, still a basuter is still an important tool in obtaining the restriction, but there are more intricate tools now. Concerning construction, for a long time there were not so many tools to make construction.

52:30 It was always more or less the same tool that Arnax used, by making unions of curves, unions of small curves and making perturbations, that was mainly the principle which was used to make construction. Maybe until recently, the combinatorial construction given by what is called now as Vero's method by gluing some pieces of curve, which is some kind of combinatorial construction, which made a very significant progress for the method of construction of algebraic curves, or more generally algebraic objects. But up to this recent development, essentially the method of construction was the one that Aramance used in this paper. What are the notions used today to cope with this statement? About the restriction part? Both. Well, in this case, construction, you still use Arnax construction. But in which language? I suppose this one, if I take the book where this statement is proved, the text, without any formula, I would be surprised. Maybe the notation could be different, but... Maybe the lines would be called L1 and L3. This is the main difference. There is an epsilon, the long term which is smaller than epsilon. Yes, I will. And the non-sigular case is not met. Well, another point, which I, yes, I forgot to mention, yes, Harvey calls me that now

55:00 if you look at survey papers or presentations, you will only find the non-singular case. There is a very important part in Arnax's paper, which is a construction way to intricate construction, which are done in the singular case. This has more or less disappeared from the modern expositions. of this. And the study, if I study on Ebert's 60s problems, there are many concerns, almost exclusively concerns, with non-singular, in non-singular varieties. So all these things on the singular case, that was surprising. I didn't know that in Arnax paper you can find the sole treatment of the singular case. What he says about the consequences, about the paste guards, is it up here somewhere? It's not, well, I don't understand what he says about space space, but I'm not talking about it. Maybe if you, even, meaning it's not... I think it's just very vague. He says you can also start studying the question of searchings of space first and that's all he said. Not much to understand again. If you take a question of a plane curve and you perturb the x-y plane and you perturb by a z You can get by non-singular curve any genus if you move in the three space. Actually, this was considered by Hilbert in his paper, that's in this paper you have curves in algebraic curves in space and so here you have

57:30 a study for instance of curves that you can draw on a quadric, okay, space, ovals of curves on the quadric, and you have also construction which are almost the same as the one by Armach, same construction for curves in space, but you are starting with curves on a fixed quadric, And you are perturbating, again, you construct a curve by going up, say, in the width here. And by perturbating the union of such curve with the intersection of a quadric with a plane and so on. It's kind of a very similar construction, which is that, visibly, Hubert was a thread, of course, on the paper, and was inspired by Armach's method for study of curves in space. When did Gilbert find that? When? When did Gilbert? This was published in Mathematical Analyse in 1891. You got the realness of the algebraical course. Maybe he had a lecture, or maybe someone told him about the construction, or maybe he would never read the paper. Maybe he never read, but he was aware, he was well aware of Armagh's construction. That's obvious. I would like to ask two clarifications of two points. still sort of sticks me in the middle of my mind, the definition of a circuit. Is it the following? The definition here in modern terms will be that a circuit is a continuous parametrization

1:00:00 by the interval of 0-1. With values in the projected plan, of course. In Armax's paper, there is absolutely no mention of what is the circuit. Exactly. That's why Armax... He refers, for Secwitz he refers to work of von Stout and Möbius, but visibly what he has in mind is not, well there can be singularities of course, he considers Secwitz singular curves, for the capy cusps, and double points, and things like that, but, so that's, if it's a continuous, let's say, well, it's an analytic, because in some of the other drawings, it's not analytic. No, it's not, it's being before you perturbed it. It's not, it's of course not that but yes but yes yeah but still there's some contradiction with what you can have a stealer to see one parameterization when you have cusps but OK, you can also have, even if you have some angles, you can have infinity parameterization. So that's not very clear. It works, but the definition is not very precise. But what is the idea? The idea is, let's say, a C1, or a continuous, or a C infinity, or an analytic, some kind of parametrization by a post-order. You know, it's a continuous image of the, well, continuous of C1 or whatever, image of the interval of 0, 1, such that the value of 0 and the value of 1 is the same, with values in the projective plane.

1:02:30 That is a, in all terms, will that be the notion of something? In all terms? With a question mark over the C1 or continuous or C infinity or whatever. That is what makes a proof of all that we call it. Well, that's what I'm asking. In modern terms, people consider non-singular case, and for non-singular case, it's easy to, well, there's no ambiguity, as simply says, and consider embeddings of the circle with the project in plain. There are two types of embedding. There are two types of embedding. There are always odd and even ones. But the odd and even is not well defined. Since why an oven circuit's oval meets always an even number of points by a little perturbation. Also if the words are the same, the meaning is different now, now some meaning is more precise, homology, homotopy and so on, of course, it's a, come se dice, si vete, one see that, but it's the spirit of the time. I'm showing me a drawing of a curve where there are two ovals, one inside the other. You take the product of two circles. Ah, like that. The union of two circles. Yes, the union of two circles and if you want it to be reducible, you perturb it. You know the way we teach about the theorem, take one curve of degree n and another one of degree n. Let the curve of degree n degenerate to n lines and the other one degenerate to n lines

1:05:00 and do the calculation. non-parallel non-parallel non-parallel you start with the union of two circles and this is the fact that when generating the number of intersection points I've long thought about 50 over 60 problems as an outsider. You should choose a definition of multiplicity to make the theorem originals. Definitions are made to have nice theorem. ...passed by one universal algorithm, and that was shown to be non-existent. And so here, over 60 problems is an infinite class of problems, and you can't solve them all, have a universal algorithm that will decide for any given integer d, for any given possible topological configuration or disposition of ovals, whether you can realize that disposition by means of such a curve should be d. And there is an algorithm like that based on Tarsus. So then the next question is, is there an efficient algorithm of that type? Some complexity class or something, and you get one that's double exponential or whatever, and that might be a way, if you had to give a yes or no answer to the question, is Silver 16 problem solvable, maybe this would be an answer. It's solvable with a certain complexity class, but it's not solvable in a worse, better place. But I have no idea what the answer It could be, it could be, but it could be easier to say, just that you count the number of, it's going to be the number of, if we need to do the product, if the general is available, it's going to be the number of, if we need to count the number of, if we need to do the So what is the objective?

1:07:30 I can do it. That would be the view. No, no, no, it would be the view. I guess, well, I don't know. Shall I do it? Then it's a single explanation. You want to do it? Because it's really, it's like it. So if you, if you, if you can, on a given curve, you can confuse the psychology, you put in on your time, and now you have to think of this, let's try it by all coefficients, so it's just one block, and it's always, and I don't know if it's one block, and I don't know if it's one block, and I don't know if it's one block, so there's no reason for it. More or less a square of the degree. So it could be, unless the number of positions is squared, there is no reason to be the main collaboration. There are more of the types of curves of GVD that you can realize that has a GVD, you see, things like connection. And then to check, and then to find all the types that you take a point in which of these connected components, in the space of our letter, and then to do it once, which I guess there's no lower bound. I guess what I would really like to see is a lower bound on the complexity of such an algorithm which would show the impossibility of solving a What construction might give you more on the number possible types of cars. you can come with the possible combination of your common relations. But I don't think of a lesson that I don't think is. There is a research where the number of pieces is really... It's really hard work, I think. Oh, yes. No, no, that's enough, I want. No, but that's not enough. It seems to me there is no more about the machine, so the only question that we don't know exactly.

1:10:00 I don't know about the number of connected components which are outside the screen. So in the parameter space, of course it doesn't say that the number of convolutional types because it permits maybe the same type in various connected components. So what moves with the novel is to take a point in another element of component and then to look at what is the topological type. And this, I think the number of convolutional is . No, but it's the indication that the accordion, if the accordion you have in mind is to take the authority in a room, but then the component of the discriminant that checks what it is, then this is . And also it depends on what type of classification you need to have. So the strongest one is rigidized, means that you can deform one curve to another without staying always in the same type, without crossing the discriminates. So, no, it's a singular case. So, is there a book where all curves of the V6 are shown? Yes, in the Soviet papers, yes. Newton goes only up to degree 5. Yes, so it's catalogued by Kochakis. Just a little remark, concerning the inequalities in the matter, I remember that there is a very short proof in Gaia, which may not be known in the paper in the 1960s. I even don't remember I don't know where it is published and how it is. I remember it's only a paper about three pages. But I can find out if you want. And I think it's written in Germany.

1:12:30 I think it is published in Proceedings of Oberwolf. It's called something like a short group of hung-ups inequality. If you want, I can find it out. I have a hope. Maybe somebody else remembers this paper. I think that there is also paper by Proof of Arnax, but I guess in the... I'm not sure. Maybe that's one thing I should look at. I think there is also proof by my client on using other types of methods. Well, starting with a curve of genius P with involution, with complex involution, and getting it this way, bound with a number of components of fixed points, so that's a real sequence. No construction backlight, no, that's fine. I think that there is a proof on, say, not curves, not blank curves, but algebraic curves of children's p in abstract, like this, and using the evolution-like conjugation to environments and more in the spirit of, say, algebraic topology. So you also mentioned the construction by zero, but apparently a black curve is also a curve which

1:15:00 If you have a very different process, then you find a very different type of curve. So you find a much more type of curve. You go hard, you go hard, you go hard, you go hard. So I'm at this position, it's supposed to be here. Thank you. I found it in another picture. It was a real problem, and one student was saying to the patient, and my question is, if you know when the patient agrees, and why is it really dangerous? The use of this notation was common, but the exponent is actually not an exponent, just to indicate the answer to three idiots. But I think that, yes, in the theory of impcience, that's something that you can commonly use. So, this notation, I don't know exactly, but it's, yes, it's very common. and it's used without further common knowledge that everybody knows what they should do.

1:17:30 Thank you. In modern presentation, for instance, in discussion about the 16th HILBAT problem, disposition of formalization, the discussion, presentation is always by degree. The question is for curve of fixed degree, I think that it was the formulation of it was this way also. Here in Arlac, the discussion is by genus. Togui is a somewhat auxiliary technical notion, but the discussion, the main important concept for the genus. And this is related to the fact that, which is explicitly mentioned in Arnax's paper, the dual aspect of curves, that sometimes you have the curve, you have also the dual curves. You have the notion of, you can consider points of a curve, or intersection of a line, for instance, with a curve, intersection points of a line with a curve. And really, you can also consider the tangent to the curve, which are issued from the balance of the plane. So that now, I mentioned specifically the duality between the two notions, the notion of degree on the one hand, the notion of class on the other hand. And so it's a duality which appears, for instance, when the first page, I don't remember exactly, when it's a piece of double point and multiple points with respect to multiple tangents, which is not a singularity, but which is a singularity of the dual curve.

1:20:00 And also, at the same time, there was a paper by Klein about Koecker's formula and formula relating the number of double points, double tangent, curves, connection points, and how many of them are real, and so on. which was published, I guess, in the same year as Arnach's paper. So that's also a trend, I think, in the algebraic geometry and study of curves in this time to have present in mind the fact that there are dual problems, about the curve and the dual curve, that's the fact that, for instance, deflection point of the curve corresponds to cusp of the dual curves, and vice versa. on the curve, with a double tantrum of the curve corresponding to double points of the dual curve, and so on. This duality was, appears here in the paper, and that's more or less was a surprise to me, importance of genus, and the genus is invariant, the duality, the genus of the curves and the genus of the dual curves are the same, from the point of view of the genus, the genus because it's a good environment when you want to have things which are invariant with respect to the singularity. Why are you surprised when you find that? Because I didn't know that it was specifically so. So I was far surprised when I saw, for instance, the first Bobay de Singularity, multiple point, respectively, multiple tangent. When I first was reading this, I was surprised. Multiple tangent singularity. And of course, because he has in mind the dual curve. The degree is the reference. The degree is the reference. This is of course related to the fact that you know the non-singular case is the one

1:22:30 which is the one. Of course it's a non-singular case for non-singular plane curves to keep the degree of the give the genus its equivalence, the genus is given by the degree. And also when you take the non-singular curve and you take its view, then that's singularity? Yes, of course. Because it has the same genus and a much bigger grip. So to get the same genus, yeah, a lot of singularity. So you have a lot of singularities to recover the essentials. And for curves in space, duality is interesting. No problem. So the dual of the projective space is made of planes, so the curves are in between, the There are tangents living in the segre variety of all the lines in three spaces, so it's complicated, it's not a projective space. That means the dual plane is the projective, but on the top it doesn't work. It's more complicated. And the duality is with the hyperplane. You can do the same for the hypersurfaces. Alright, let's switch to the cockpit.

1:25:00 We started the second part of the meeting. The paper of Cartan is divided in two parts. The second part of Cartan's paper is different from the first one. In the first one, Cartan establishes the mine results, mine maybe for this time, the mine results for these objects, and in the second part, Cartan studies the structure of the objects. In order to have a time of discussion, we examine only the result obtained. Before starting from the second part, I believe it is better to recall some definitions and some notations used by Kata. What do you mean by second part? The second part is from the section 8. In the first part, in my opinion, and the part means also where we have arrived at the end of section 7, in the first section. But I begin with recording the notation and definition in section 6. In the English or French version it's exactly the same, the number of propositions and the number of theorem are the same and the number of sections, the numbers of

1:27:30 sessions are the same. So, Kattan denotes by, let, let be an analytic manifold. notation of character air is the shift of gems of analytic a real analytic manifold, so is the shift of real, the germ of real analytic function. and Catan defines an analytic real, analytic subset, Locary. Locary means exactly in the same way as in the complex case. A is a real analytic subset if for any if A is closed, and for any point of A, there exists an equal U of A, such that U intersection A is described by finite, finite remaining functions, analytic functions, Fk, zero, where when the functions fi are analytic in u. The definition is local and is exactly the same definition as an analytic object in an analytic subspace of Cien. Catin remarks also that if the local description has a Jacobian of

1:30:00 maximum rank, this is exactly the definition of analytics sub-handifold of Pooh. Kertan defines also the sheath attached to R in this way. Take a point X in V. If X is in A, the shift XX is the shift of germs of all germs in LXU, vanishing at x, not x, x in two, if x is enough, if a germ vanishes in a point vanishes in a neighborhood. If x is not enough, if x is at two. This is the definition, and if E is coherent, then Cartan calls A coherent. This is a definition. A is coherent if the shift of idea of all germs vanishing Qatar is coherent. This is the definition. Qatar remarks that the air boom, the shift of all analytic function is coherent, but from a theorem very similar to the theorem of Oka,

1:32:30 but remarks also that he can be not coherent, not as coherent, can be not coherent. And remarks this in an example, the very celebrated Cartan umbrella in another part. At this point, Cartan establishes the two theorems, or the one theorem as you prefer, A and B. So this and some consequence of the theorems conclude the first part. Why the first part? The second part is dedicated to describe, define also, the structure of the new objects, the real analytic sense. I believe that there are people who don't understand the notion of coherence. Ah, I can repeat the notion of coherence? Alright, what is coherence? Maybe to say a few words on the shift, not the shift, but you are the only one.