 You speak on F-signature of some hypersurfaces. OK, so thank you, Mark. So first of all, let me thank the organizers for inviting me here. It's a great pleasure for me to be here and speak in front of such a big and great audience of mathematicians of many generations. And yeah, thank you very much. So the work I want to present today is F-signature of some hypersurfaces. And it is a joint work with Francesco Zermann and PhD student in general. So let me start by fixing some notation. We will work over A, which is the power series ring in S variables. You will use the letter S to denote the number of variables. I want to keep N to be the argument of the functions we will consider. And I will assume the field K is perfect of positive characteristic P. Then we consider an ideal I inside A. And we will work over the quotient R, R mod I. And I will denote by M the maximal ideal in R. So the function I'm going to define can be defined in much more general setting. But since the results will be here, so we will restrict from this from the beginning. So we'll start with introduction. And we start with the Hilbert-Kuhn's function, which I will denote by HKR of N, where N is a positive integer. And this is nothing but the length of the quotient of R modulo de Frobenius powers of M P to the N. And this is the Hilbert-Kuhn's function, which has been already introduced in previous talks, and has been studied first by Kuhn's. We took as a model the Hilbert-Sommer function, where here we take ordinary powers of the ideal. Instead, here we take the Frobenius powers. So M P square brackets P to the N is the ideal generated, but all powers of P to the N of the element of the maximal ideal. And was Monski who gave to this function the name Hilbert-Kuhn's? And also, Monski proved that asymptotically it grows with the leading term, which is polynomial of degree D in P, where D is the dimension of the ring. So there's a leading coefficient EHKR P D N plus a term, which is big O of P D minus 1 N. And the leading coefficient is called the Hilbert-Kuhn's multiplicity of the ring N. It's a real number, which is always greater or equal than 1. But one can do a little bit better than this if we, for example, assume that R is also regular in co-dimension 1. Then not only we have a leading coefficient, but we have a second term. So EHKR N Hilbert-Kuhn's D to the N plus beta, where some real number beta, P to the D minus 1 N, plus big O of P D minus 2. And this is due to unicamic Dermot and Monski in the normal situation, and later generalize to regular in co-dimension 1 by Chan and Kurano. And there are examples by Han and Monski that prove that one cannot do better. So in general, we cannot expect a third coefficient to exist. But I will come back to these examples in a moment. First, I want to introduce the second big player, or actually the main big player of this talk, which is the F-signature function, which I will denote by FS R of N. And this is nothing but the free rank over R of the module R1 over P to the N. So I denote by R1 over P to the N. The over ring of R, where I join P and roots of unity. And the free rank is the maximum rank of a free direct summand of R. So free rank is the maximum rank of a free R summand. And this was introduced by Smith, or studied first by Smith and van der Berg in the context of ring of finite F representation type. Then Unicay and Loichke studied it further in greater generality, and we were interested in the leading coefficient, which I will define in a moment. And it's worth mentioning that all around the same time, also Watanabe Yoshida introduced the notion of minimal Hilbert-Kunz multiplicity, which is also closely related to this. And also for the F-signature function, it is known by work of Tucker that it has a leading coefficient denoted by SR, which is called F-signature of the ring. And I think I never written anywhere that D, sorry, I said it, but I didn't write it, is the dimension of the ring. So plus a term, which is big O of P to the D minus 1, N. SR is a real number, but differently from the Hilbert-Kunz lies in the interval 0 and 1. Much of the effort so far, I've been devoted to the study of this invariant, I think, last week in the school. There have been plenty of time to speak about them. And they are very important, because in some sense, they characterize the singularities of the ring. It is known that under mild assumption, both the Hilbert-Kunz and F-signature are one if and only if the ring is regular. And in general, the idea is that the higher the Hilbert-Kunz, the more complicated the singularities, and the closer to 0, the F-signature, the more complicated. Let me mention that differently from the Hilbert-Kunz, which is always positive, the F-signature can be 0, and it is known by work of Albert Bach and Loiske that it is 0 precisely when the ring is not strongly regular. But the focus for me today is not on the leading coefficient, but more on the function. Because here I wrote a big O term of this, and also here, also, let me say that also for the F-signature, in many cases, the second coefficient is known. But in general, the shape of the function can be more complicated. And it's not clear at all, although many examples are known, and here I just wrote a sample of them, just to mention how the situation goes. For example, for stylizing ring and binomial hyper surfaces, the Hilbert-Kunz function is known by work of Konka. Then for projective curves, it is known by work of Trivedi and Brenner. And also for normal affine, semi-group rings, this is by the work of Brunz and Watanabe. And then let me also mention 2 by 2 minors of generic matrices. This is also known by work of Miller, Robinson, and Swanson. Concerning the F-signature, the shape and the precise behavior of the function is known for normal affine semi-group rings by work of von Karff and Singh, and for invariant rings by work of myself and the Stephanie. And for invariant rings, it is known by work of Brinkmann, but only the Hilbert-Kunz, only for G in dimension 2 subgroup of SL2 case, so only in this special case. There are some more examples, for example, full-frag varieties by work of Fakrudin and Trivedi and some other cases, but not so many. There's still a big class I left out that will come to it in a moment. What is common to all these examples that in all these cases, the function at Hilbert-Kunz of F-signature are quasi-polynomial. So what I mean by this, I mean that you can write them as a sum of A i n, P i to the n, where i n, the coefficients, are periodic functions. Or eventually, they might be constant. In some cases, the function is even a polynomial like the classical case, the Hilbert sum. And of course, the leading coefficient is known to be constant because it's the Hilbert-Kunz or the F-signature respectively. And these cases are known, and while there is, of course, a deep connection between the two functions, it's not clear that the method that will apply to one will apply to the others. For example, I don't know how the geometric methods developed by Trivedi and Brenner could be applied to the F-signature function and also the method that Alessandro and I study here. I don't know if they will extend to the Hilbert-Kunz. But still, as I was saying before, there is a class of example I didn't mention, which is a class of example where the function is not quasi-polynomial in P to the n, and these are due to An and Monsky, and later by Monsky and Teixeira, and they produce the example of non-quasi-polynomial Hilbert-Kunz function. And with this, we come to the second part of the talk, which is P fractals, where I will try to briefly explain what are the methods of An, Monsky, and later by Monsky and Teixeira for the Hilbert-Kunz and out, they can be applied to the F-signature. So from now on, we restrict to the case of hyper-surfaces. So I is now generated by a single element. Okay, so let me add this to the notation. Now I is, in case I raise just hyper-surface, and the idea of An and Monsky and then Monsky and Teixeira is to associate to the hyper-surface as a function, which we denote phi F and takes value in a special set of this form, P to the n, and this is one over P to the Sn times the length of A modulo, the maximal ideal, probenius powers, and F to the A. Okay, I think I can erase the table now. Okay, so the idea is to associate to any hyper-surface a function of this form, let me close the parenthesis, and this phi F is a function from I to Q where I consists of elements of the form I divided P to the M, where A is an integer, and also M, natural numbers, intersection with the interval zero, one. So this is nothing but, if you forget about this factor, this is nothing but somehow the Hilbert-Kuhn's function of the powers of F, F to the A. In fact, for A equal one, you recover precisely the Hilbert-Kuhn's function despite this normalization factor. But the idea is that if you are able to collect information altogether of this function, maybe you can get some information on the Hilbert-Kuhn's. Yes. No, S is the number of variables here. So A is still the power series. So yeah, it's the dimension plus one, if you want. That's, okay, so far so good. So then they introduce operators of these four. So for any natural number N and any B between zero and P to the N, they define an operator T P to the N B of phi of T P to the M to be the inappropriate translation of the function phi T plus B divided P to the N plus M for any function phi in Q to the I. So it's kind of translation of the function and they define Monsky and Teixeira. They say that the function phi is a peak fractal. This T P N, yes. Yes, sure. Yeah, see. Yes. It's a natural number. So the index set. Any natural number. Any natural number. So it takes the domain is this I which consists of fraction of the form I divided P to the M. So the denominator should be a power of P and the numerator any natural number A, just like that. And so you, and then you define a function in this way. Given F, the value of this function in the fraction AP to the M is this number. So the length, you hear the maximal idea that A gives you the power of the F to the A's. Sorry, maybe it's too small. Yeah, sorry. That's F to the A. So A plays a role in the sense that controls the power of the upper surface. Sorry, it was too small. Oh yes, I'm so used to write length over R that yeah, that I do it automatically. Thank you. So, and they say that the function in general a function in this space from I to Q but one of these form is a P fractal is whenever you apply all possible operators of this four for every N and every B, then this spends a finite dimensional subspace of the space of functions, okay? So, and the name prefractal comes from the fact that if you impose this condition, you can see this function phi from the interval zero one while a subset of the interval zero one to the real positive numbers. And if you assume this prefractality condition, then the graph of the function will have a shape that is kind of a fractal. So that's the way of the name. It will repeat itself because the idea is that the function is given only by a finite number of information. And in fact, one of the main results of Monsky and Teixeira is that if phi F is a P fractal, then the Hilbert-Kunds series in the sense of the generating series of a modulo F is rational. And this implies in particular that the Hilbert-Kunds multiplicity is rational. And also, moreover, if you know more information about the relation of this function, you can also try to compute the function more explicitly. So, the first thing we did with Francesco is to define a weakening of this condition. So we say that phi is a weak prefractal if only a subspace of these operators, so we consider only operators of the form tpn zero with no b, span a finite dimensional vector space. So, and this has a price in the sense that if you only assume this, then the shape of the function is not anymore so nice. You lose this fractality. But still this is, and this is our first result, is that in this case, this condition is sufficient and necessary for the Hilbert-Kunds to be rational. So phi F is a weak prefractal if and only if the generating series of the Hilbert-Kunds is rational. So a quotient of two polynomials. So, and this is the first statement. The second statement is that if we consider the function phi tilde, which I will introduce in a moment, then this is a weak prefractal if and only if the generating series of the F-signature function is rational. Tell you what is phi tilde. Phi tilde is the reflection of phi. So phi tilde in a value t is phi of one minus t. So phi tilde of t is this phi of one minus t. This is called the reflection. Phi F in this case, well, anywhere. So in some sense, what does this tell us? Apart from that we have a characterization of the rationality of the series, but I mean, is this useful? So as Trivedi already pointed out, one question is, well, but does this help you to compute? Well, the answer is yes, because in some sense this function phi F, when it approaches to zero, it gives you information about the Hilbert croons and when it approaches the value one, it gives you information about the F-signature. And so you may try to use these techniques of prefractal to give actual computation of the F-signature. And this we'll do in the third part of the talk. And as a test, we focus first on Fermat hypersurfaces because in this case there were already some open questions that we wanted to try to answer. So let me now fix the notation for the last part of the talk. We can see the Fermat hypersurfaces. So ADS is now the Fermat hypersurface in S variables and of degree D. So I will denoted this by ADS. So now the second here, well, K is still perfect, but we just reduce it a little bit. So but A is still that one. Okay, so what is known about this? So first of all, the F-purity is known by Oxtar and Roberts in the famous paper that was already mentioned. So we know that there are some cases we know what happens. So if D is bigger than S or P is smaller or equal than D, then ADS is not F-pure. So, and this for the F-signature function means that the F-signature is identically zero, so not much interesting. In the case D equal S, then ADS is F-pure if and only if P is congruent to one, modulo D. And in this case, one can compute that the F-signature is identically one, so also not so much interesting. In particular, the ring is not strongly F-regular. Now we come to the most interesting situation. This is identically one dysfunction for all N. The function, the F-signature function, not the leading coefficient, the leading coefficient is zero. Sorry, it means that, I mean, actually it means that they always have one splitting for every, somehow, for every N in this sense. Yeah, the leading coefficient is zero. So in the case when D is bigger than S, then it's known that ADS, the Fermat, is strongly F-regular, so the F-signature, the limit is the leading coefficient is down zero. Strongly F-rational for P large enough. Yeah, I mean irregular, sorry, strongly F-regular. Sorry, got confused. Okay, so what we can say in this case, yeah, yeah, yeah. Sorry. So what we can say in this case, so let's assume and then also D is at least two, P is greater than D and we have a technical assumption, so P is congruent to plus or minus one mod D, and we have that either S, P minus one D is even in the case that P is congruent to one modulo D or S times P plus one divided by D is even in the case that P is congruent to minus one mod D, then there exist two integers B and C, where B is between zero and P to the S minus three, S is the number of variables, such that the F-signature, another chalk, F-signature function in this case is the F-signature times P to the S minus one N, and this was known, plus one minus the F-signature times B to the N and this is true for all N. So we only have two terms in the function, the leading part which we know and the part which is of the order P, S to the S minus three, so the dimension minus two because S is the number of variables, so there exists a second coefficient which is zero, but when B is bigger than one or it's not the power of P, there are no more coefficients, so this is not quasi polynomial in particular, and we also know in this case that the F-signature is given by minus C P to the S minus one minus B, in particular, it is a rational number, so far so good. So people may want to know, okay, but can you tell something more precise about this value? Well, yes, even in the case S equal to D plus one, which is the, yeah, the first, let's say interesting case, there's a theorem by Watanabe and Yoshida in 2004, they prove that the X-signature, sorry, the limit, the leading coefficient is more or less equal than one divided by two to the D minus one factorial and they ask whether equality holds, and for D equal to, this is, the answer is yes, it is known that this value is one half and the F-signature, I mean, we get an F-signature of a quadric in three variables. This is known by many ways that this is true, but for D equal three, we could compute the precise value of S, A, three, four, and this is three p, p minus one, p plus one divided by eight, three p third, minus p plus two, minus one to the alpha plus one, so very complicated number, which depends on p where alpha is either zero in the case that p is congruent to one, mod D or one if p is congruent to two, mod D, but in both cases, this value is strictly smaller than one eighth, which is this value for D equal three, so the answer for D equal three is no, but you may notice that still, if you take the limit for p going to infinity, you get exactly these three cancels, you get one eighth, so in some sense, the bound of Watanabe Yoshida is optimal in this sense, so still, the value is smaller than one eighth, but at infinity, it goes to one eighth, and I think this is all I wanted to say and thank you for your attention. Well, thank you very much. Is there some questions? Or if not, thank you again. Thank you.