 It's an asymptotic resurgence of homogeneous ideals. Thank you, Dave. Thank you, the organizers, for having this conference here, inviting me to give a talk. This is the first international conference that I am attending post-COVID. So it's meeting a lot of people after a long time. It was really exciting to be here. It was still exciting to be here. And I clearly remember the last time I was in ICTP. That was in 2004. I was just an year after my graduation. And I remember being very, very nervous giving a talk here with the audience where full of people whose papers I read and grew as a community algebraist. So it was like, Huneke was there, Tito Wala, Wes Kinsellors, Marilena was all people. I was really nervous. At that time, I clearly remember those days. That was, I think, held in Adriatico, that time. Anyway, so thanks once again. So let me begin my talk. This is a joint work with Arumind Kumar and Vivek Mukunddin. So the basic problem is what is known as containment problem, that given an ideal, how are the containment relation between the symbolic powers and the ordinary powers? So symbolic powers, I think, we already have. It's been already introduced in the series of talks. So I just ISR localized at P intersection with R. P varies over associated primes. So the problem is given a T, find the least S such that I symbolic S is contained in IT. So well, the very generally stated problem, how do you answer this? So there was a lot of attempts in this direction. So with a series of papers, starting with Irina Swanson, then I and Lazaz fell the Smith, Hawks to Huneke. I think these two papers, I think all these papers have already been mentioned in a couple of talks here. And this is the most recent, I think, in a quick characteristic case. So one has that i to the power ht, symbolic ht, is contained in i to the power t, where h denotes the big height. Big height is the maximal height of associated primes. That's called the big height. So with this, Huneke asked this question, can you refine this in some good cases? So for example, this is a particular question that Huneke has raised. If P is a height to prime ideal in a regular local ring of dimension 3. So the earlier theorem says that P symbolic 4 is contained in P2. But he asked whether P symbolic 3 is contained in P square. And this question is wide open. It's, I think this is a couple of decades old problem, but it's known in only very few cases. Very recently, there is this proof by Griffo in the case of space monomial curves. And in general, this is very wide open. It will be interesting to see some good classes of prime ideals satisfying this, I mean, affirmatively answering this question. So I mean, taking a cue from this theorem, Harburn asked a more general question that given a radical ideal in a regular ring, do we have a finer containment in general? Like, look at this i to the power symbolic, ht minus h plus 1. Is it contained in i to the power t? Here, h is the big height. Whether this is true for all t, this is not true. There are counter examples for this one. But yes, this holds true for some classes. It's known for several classes of ideals. But then Griffo refined this question and asked whether such an inclusion holds true for large values of t, for all large values of t, whether we know that such a containment holds. And that is, of course, this is a very recent question. And there are some positive answers. So far, there are no counter example to this question. So in understanding this containment problem, Bochian-Harburn introduces an invariant called resurgence of an ideal i. So resurgence is defined as rho of i equal to the supreme of s by t, where i s is not contained in i t. So if you take a rational number, which is, OK. So the corollary to the theorem that was earlier theorem is that rho, if I take an ideal in a regular ring, radical ideal in a regular ring, rho i is less than or equal to the big height. So the importance of this rho i is that if you take a rational number s by t, which is bigger than rho i, the resurgence, then that will ensure this containment. So rho i is related to the containment in this way, resurgence. So once you know the resurgence, you know the containment problem. So the idea is to compute resurgence. Of course, it's a very difficult problem. Computing resurgence is very hard. So in studying resurgence, Guadal, Harburn, and Van Thuyl introduced another invariant, which is called asymptotic resurgence of i. So they defined it like this, s by t, where i r, OK, this should be, i r s is not contained in i t for all large values of r. So one can, it's very easy to see that the resurgence is bounded below or asymptotic resurgence is bounded above by the resurgence. They, in fact, proved that this is such a sequence of inequalities. This is fairly obvious. And alpha i is the minimal degree of a minimal generator of i. And alpha hat i is the Waldschmidt constant. So they proved this inequalities. And then now the question is whether, how do you compute? How do you get an upper bound, get a lower bound for the asymptotic resurgence, for the resurgence, and see whether compute classes where you can actually find classes where you can actually compute the resurgence and asymptotic resurgence. So Griffo and then, with Griffo, Heunekin, Mukundin, they showed that the stable carbon conjecture is true if rho i or the asymptotic resurgence number, either the resurgence or the asymptotic resurgence is strictly less than a big height of i. So in the paper with Griffo, Heunekin, Mukundin, when this happens, they call it expected resurgence. So when an ideal has expected resurgence, the stable carbon conjecture is true. One can state this result in that way. Now coming back to the computation of resurgence and asymptotic resurgence, one can ask, how do you compute this? It's kind of stated in a supremum set of rational numbers. How do you compute this? It's really tough. One question would be to see whether you can reduce it to finite steps. So this is, Deepaskal and Drabkin proved that if this holds true, then yes. One can use certain methods to compute rho i in the sense that you can use, you can say that this comes from a finite set. The resurgence can be computed using that set. You can reduce it to a finite set. So the ideas, if the symbolic resulgebra is noetherian, then in that case, rho i has expected resurgence and therefore one can prove that this is a rational number. See, that's the corollary of this Deepaskal-Drabkin result that then it becomes a maximum of a finite set. Therefore, it's a rational number. In general, it's not known. It's not proved otherwise, but it is not known whether it's a rational number, the resurgence. So we were trying to explore this. So first thing that we did was to look at the, trying to see if we can get some nice upper bound for some nice classes. So this is the first result that we have in this direction is a generalization of Griffo, Hineke, Mukundan theorem. So they prove this in a more restricted setup. S is a regular local ring of dimension three with some, with very similar hypothesis. But we basically observed that the techniques do not really require all those restricted hypotheses. And we could prove this result that we if the ideal satisfies some nice conditions, then one can obtain a very sharp upper bound. Very sharp. P, oh, yeah, I'm sorry. Yeah, P prime, yes. There exists a prime ideal P such that this happens. So this is a very sharp upper bound in the sense that we have some nice infinite class of ideals where this is achieved. Now in this direction, when we talk about the resurgence containment problem, whenever we talk about examples to show that something is true, something is not true, there are two types of examples that exist, one coming from the geometry and one coming from the combinatorial side. So the geometrical examples are mainly, I mean this, I think, Fermat's curve and then some points in projective plane. So it comes from that side. And the combinatorial side comes from, this is precisely, I work in this side of commutative algebra on the combinatorial side. So all these, everything that I'll be dealing within this talk will be, examples will be combinatorial. This is one such. So this is what is called edge ideal of odd cycle, cycle of length 2n minus 1. So in such case, we can compute the resurgence and it is equal to this one. So it'll turn out to be equal to, this k will be equal to 1 and precisely that. So in this case, the maximal ideal will act as the prime ideal required here. And then while dealing with this, so we were looking for more examples or more ways of approaching the resurgence. So what we saw is that, so in my paper with Thay and Sankhneel and Abu, we looked at the resurgence of some of ideals where these ideals are generated in distinct variables. So here, we are looking at what happens to the resurgence of the product. We also have a very similar result in the case of intersection with slightly more restricted hypothesis. But if you take inj to be non-zero proper ideals generated in this joint set of variables. So one is generated by x1 up to xn. The other is generated by y1 up to ym. I mean, using these variables, polynomials and this homogeneous polynomials in these variables. Then the resurgence of the product is maximum of the resurgence of these two. And the same holds true in the case of asymptotic resurgence as well. And as I mentioned, we proved certain bounds for the resurgence in this paper with Thay, Ha, Sankhneel, and Abu. So here, we wanted to see what happens in nice situations. We have two bounds. We have given upper bound and lower bound. And if the symbolic powers of one of them coincide with the ordinary power, then this rho of i plus j is equal to the asymptotic resurgence of the sum is equal to the resurgence of i plus j, same as resurgence of the other ideal, second ideal. And this is basically used as a tool in the more general situation if you have several ideals. So each of them generated in distinct variables. And we are assuming that rho is equal for 1 to k of them. We are not really assuming that they are equal. We are only assuming that rho is equal to 1. So here, the resurgence is equal to 1, not necessarily imply that the symbolic and ordinary powers are the same. Now, we just have the supremum to be equal to 1. It need not necessarily imply. So here, when I think this crept in from here, curtain paste error. Apologize for that. I'm sorry. So this is not there. Then for all j from 1, sorry. Then this statement, please delete. The resurgence of i1 up to ik is maximum of these numbers. So yeah, so this is, we encountered all these when we were looking at resurgence of square free monomial ideals. That's how we try to look at this one and then realize that to prove many of these results, we don't need square free or homogeneous and several things. So we were looking at this result by De Pascal and Drabkin. If I look at an ideal in a polynomial ring and ideal is a square free monomial ideal of big height h, then they proved that the asymptotic resurgence is less than or equal to 1 by L. L is the number of variables. So this can be a large bound. So what we have shown is that if i is a square free monomial ideal of big height h, then we have a containment i symbolic hr minus h is contained in ir for all r bigger than or equal to chi of i. So this is an invariant that comes from the combinatorics. This is called what is called a chromatic number. So this asymptotic resurgence is less than or equal to h minus 1 by chromatic number. So what is exactly chromatic number? Let me give a quick. So if you have a graph, let's say, so a coloring is an assignment of colors. A proper coloring is an assignment of colors to the vertices such that no edge is monochromatic in the sense both sides of the edge will have different colors. So the minimum number required to color such a do a proper coloring is called the chromatic number. And of course, there is what are called edge ideals. So let me quickly define what is an edge ideal. If I have this x1, x2, x3, x4, a graph on forward vertices, then I define this ideal x1, x2, x1, x2, x3, x3, x4, and x4, x1. This in the polynomial ring. I just fix a field k and look at the ideal. So this was first defined by Willa-Rail in 1990. And this, I mean, a lot of study has gone into understanding the correspondence between the algebra of this ideal and the combinatorics of the corresponding graph. So here, when we use this chromatic number of this graph, so there is this concept of hypergraph. Here, each edge will be represented by a pair of vertices. In a hypergraph, it can be a sub-collection of vertices. It need not necessarily be some, it need not necessarily be two. So in that case, there is the definition of chromatic number, et cetera, and it makes sense there. And it can also be seen that, or it can be proved that a cover ideal, any square free monomial ideal can be expressed as a cover ideal of a hypergraph. So therefore, we have converted this problem into a combinatorial setup. And there, using the combinatorics, we have found a much finer upper bound. So this can be chi of i, the chromatic number. If the graph is what is called a star graph, n vertices, and there is a vertex at the center, then you can assign one color here. And all these colors can be second color. That is a proper coloring. And therefore, you need only two colors to color such a graph, even if you have any number of vertices. So therefore, in such a case, this will be 2. So it'll be h minus 1 by 2. Our bound will be h minus 1 by 2, while this will be h minus 1 by n. So that way, this is a much finer upper bound. One can use some techniques to get an upper bound or use this to get an upper bound for the resurgence as well. But it requires some work. So this is the edge ideal, definition of edge ideals. And the cover ideal, this is the main part of our talk. What is called a cover ideal? A vertex cover is a collection of vertices as a vertex cover if it intersects with every edge of the graph. And for each vertex cover, look at the product of all those variables, and take the monomial, and look at the monomial ideal generated by those monomials. And that's called the cover ideal of the graph G. So these are nice classes. Cover ideals are radical, height to un-mixed ideals. So it's a good class to check these hard conjecture, hard problems. You can try to see if they are true in this case, or you can do something in this class. So what we have, the first step, four minutes I have. So the first step is that we have, this is again, we're answering a question of Griffo. We have obtained some, again, using chromatic numbers. We have obtained containment relation. And this is very nice bounds for the resurgence and asymptotic resurgence. So given a graph G, omega G is the size of the largest complete graph. A click is the complete graph inside G, size of the largest complete graph inside G. And this is alpha G is the maximum size of an independent set. Independent set is a set where, you know, set of vertices where there are no edges between them. So if you have these two, then one can bound the row and asymptotic resurgence with these two. So this is a, I mean, philosophically, it's a beautiful bound in the sense that the resurgence and asymptotic resurgence define a very, very purely algebraic way. But the bounds are purely combinatorial. It's just, you know, you're bounding it using combinatorial invariance. So this gives another interaction between the algebraic invariance of the ideal and the combinatorial invariance of the graph. And certain perfect graphs where, you know, perfect graph is precisely the sky G equal to omega G. In that case, we can bound, we can get the exact, this is a large class. And in that, for that class, we have the resurgence and asymptotic resurgence. And while we were writing up or, you know, I think we had already, after writing it up and putting it in on archive, Villarreal contacted us saying that, you know, they also have proved, but of course using very different techniques. They have obtained this lower bound, actually. And then, of course, the corollary. So this is, you know, again, our exploration continued in the case of edge cover ideals. So they have proved that G is bipartite, if and only if J, G power S is, these two are equal for all S. We, in fact, generalize it to say that this is equal, I mean, rho and asymptotic resurgence both are one, if and only if, I mean, it is equivalent to saying that G is bipartite. So we, I think I'll rush through some of the results. So we have some more computations of resurgence and asymptotic resurgence. And then, we have few more results in the case of edge ideals, which again, you know, this is again a result of Simis-Vasconcelis Villarreal, where they say that it's bipartite, if and only if these two coincide, but we have it for in the more general situation. And we compute resurgence of edge ideals in some cases. So I would like to just conclude with some questions. In the sense, what are the some open questions in this direction? One is the first one that I discussed, in the case question, whether given a height to prime ideal in a three-dimensional local ring is the third symbolic power contained in P square. And then the stable Haber-Konjecture is still wide open. And identify classes of ideals for which stable Haber-Konjecture is true. And then we have proved several results for the edge ideals, cover ideals of graphs. One can think of generalizing this to the hypergraphs so that it will be, you know, it is valid for more general class of square free monomial ideals. And then bounds for resurgence and asymptotic resurgence for edge ideals. We have mainly got it for cover ideals, similarly for edge ideals. With that conclude, thank you for your attention. Are there questions? Yeah. This time, let me give you the microphones. Yeah, but I guess the point is that anybody on the listening remotely can't hear it. Sorry. Jayanthan, somewhat tangentially related question for either edge ideal or cover ideal, do you think we have any hope of describing symbolic powers, not just for lower powers, but symbolic powers in terms of the graph? For edge ideals, it is described for some classes, for example, cycles. But I'm talking about in general. In general, for square it is, I know, second symbolic power. Yeah, yeah. No, but as far as I know, it will be tough. I mean, of course we are looking for the non-bipartite case because bipartite, all these are same. In the non-bipartite case, I think unicyclic graphs, I think there is a description. For the non-unicyclic graph, For unicyclic, you mean just one cycle? One cycle, so one odd cycle along with a lot of trees attached to several vertices. So for such graphs, the edge ideal, the symbolic powers are described. But if there are two cycles, I doubt, and even in the bicyclic case, I doubt there is a clear description. What about cover ideals? Any description there? Well, cover ideals, for the case of symbolic powers, cover ideals are slightly better in the sense that they are, so for example, some of the results that we have proved. I mean, if you can decompose it into two graphs, that can give you some intersection to cells and so on. If there is, you know, you can write it as a click sum of two graphs, that can give you some. So cover ideals, I have better hope, but edge ideals are, in terms of symbolic powers, pretty hard. Okay, well, let's thank Jayathan again.