 All right, so with that taking care of me pass the mic and video on to Pearson for the last talk of the session. And I'd like us all to give a huge round of applause to mark as well as all those people. Both at PCMI and you. So so let's let's unmute and give a big clap. I've originally been having an enormous amount of fun. Try not to blow it with the last talk. And let's let's keep the fun. We were discussing an enrichment of degree and let's go back to this this topological degree so we have homotopy classes of pointed maps from the topological and spirit to itself and this roughly counts the number of pre images. And more specifically with a sign. So given a map. Its degree is the sum of local degrees, and that can be computed at any point in the target. So PN. Let's assume that the inverse image is a discrete set of points. One through Q. Capital one. And we have that the degree of the map is the sum of the local degrees at the at the inverse image points. And the local degree. Can be computed with the Jacobian. Great QF can be computed. Pick local coordinates. The orientational compatible, the local coordinates. Near QI. What. Why one through why and. The local coordinates in your P. Compatible with an orientation chosen on us. Compatible. With an orientation. Once we have local coordinates, then F is a N functions of the X. And we can form the Jacobian. So we have, let's call the Jacobian J. The Jacobian J is the determinant of the partial of Fi with respect to XJ. Over this matrix. And the degree. Is plus one. If J at QI is positive, so J. So F preserves the orientation and minus one. If J at QI is negative and we ended with, with a couple of questions. And one question is what if the zeros of that for not multiplicity one. Another way to say that is what happens. If J is equal to zero. And so there's a beautiful. Answer to this. Given by the Eisenberg. Levine that this one is Harold Levine. And this one is actually signature formula in the 1970s. The late 1970s, maybe around 1980. So it says that if we want to know the local degree, we can make a bilinear form and take its, its signature. So this is the signature. Remember, for a bilinear form over R, we had, we could diagonalize it with plus ones and minus ones on the diagonal and the signature is the number of plus ones. So let's define a bilinear form called Omega Eisenberg Levine from Sheffield. Over R. And then we'll have that our answer to the local degree is, is given by the square drive form or bilinear form. Where Omega EKL is the isomorphism class of the following bilinear form will form the quotient. So, we're going to generalize from the real numbers to an arbitrary field K so let's let's prepare for doing that let's let K denote R and then in a second we'll just let K be a few. We're going to let Q be the localization of our variables and let's, let's work it. This is the local ring at X so X corresponds to some ideal and then there's a ring ring right there. And then over our coordinates of our functions F1 through FN. And the fact that X was an isolated zero here. Where did it go. It gives us that this is finite dimensional. Q is a finite dimensional local complete intersection. And there's a lot of beautiful duality theory, sort of the stair duality coherent duality. And one of the things that this this theory gives is a whole class of rings which are self dual. So, Gorinstein refers to when your your your dual or your dualizing sheath is locally free. And so this gives us that the harm from from from Q to to K or R is isomorphic to Q to get a form. We need an isomorphism that'll give a form functorily but we want to distinguish one so better there's a canonical or a distinguished isomorphism. Coming, basically from the Jacobian, but the Jacobian insert positive characteristic, let's say from a distinguished, it's called a soccer and commutative algebra, soccer element, and there are papers by Sheja and Stork giving lovely results on this, and this gives us a bilinear form. We make this so explicit that you can stick it into a computer. Indeed Sabrina Pauley has, and how does it go into a computer goes like so explicitly. We still have our Jacobian as our determinant. And now this is can be viewed as a function in our ring. This is in in Q. So we can choose any linear map. Let's call it Ada Q to K. Okay, linear. Such that data of the Jacobian goes to the dimension. And then the isomorphism class. Say the characteristic is not to have this the associated and I didn't quite give you the form so that that's then omega Ekl from Q cross Q to K it's a bilinear form on this finite dimensional vector space Q it's also a ring takes two functions omega Ekl of two functions g h is Ada of the product g times h and then the isomorphism class doesn't depend on our choice of Ada. So an example that that will will use in a minute. If we had just f in one variable, so a one to a one or R one to R one, and we had f of z is equal to z squared, then Q is K x and we're at the point zero, localized at zero over x squared, which is just K x over x squared, the Jacobian is to X. And we have a basis one x, we can make a grand matrix associated to the bilinear form, Ekl omega Ekl. X times X is zero. G h or G is equal to X and X equals X is zero so there's a zero here. And our Jacobian is is basically X it's to X and it has to get sent to to. Let's make the characteristic not to here as well. Whenever. This is divisible by the characteristic we need to pass that distinguished shock a moment. And as it's a canonical way to write that thanks to Shaysha and short. And then we didn't specify what this was we could choose it to be zero it doesn't matter. By, by changing a basis by here. So this is in the generators we had last time, which were the rank one forms. This is our hyperbolic element. Um, so, Eisenberg asked. What about over an arbitrary field. So, he notes that this bilinear form is defined over an arbitrary field. Okay, of characteristic, not to. And as a degree. And does this notion of degree have some sort of topological or homological interpretation. The answer is yes, it was the degree from a one homotopy theory about 20 years before, before a one homotopy theory. So, in joint work with Jesse Cass this form is identified as the local degree in GWK. And we proved this for, for Q a rational point. And then work of razzleton and Birkland, and the keen and Montaro and Morgan opi, and I see at least those two folks right here. Consider the case, take care of the case when this is separable with the canonical map from K to the residue field is an isomorphism. So the in terms of real local degrees, we have a quadratic form showing up and a one homotopy theory also makes a quadratic or bilinear form show up. I had a question in the chat last time about why bilinear forms I have the same question myself but there are there are some comments. While we're here, let's take a detour on a one number numbers. What the characteristic of a ground field K to be to be not equal to two. And that there are a bunch of pretty points of you on Miller numbers and I like this article of Orlik that there's a it's an old article but there's there's a link in the notes. Let's start with one, the simplest kind of singularity. I'm going to show some simplest singularity. And a singularity is something that looks like this, this, this is a node. This is something that looks like that. And let's define it. So a node is defined. Over K bar, and I'll break the closed field to be a point P and X with its local ring completed that looks like a standard node. And you can get rid of it. So in the plane, it looks like it looks like so. If you start with a more complicated singularity, and move out a little bit, you're going to watch that singularity break up into the simplest singularities are the nodes. Let's give ourselves a hyper surface singularity P. We'll call it hyper surface X. And if you vary X in a family. So, let's define X T to be, we start with our polynomial F, we're going to add an error term for some fixed a one a two. And X N equals T you can make these families more general but this one has a nice tautological way of looking at it as a degree. So, then the potentially more complicated singularity P will bifurcate into nodes. The number of nodes is this Miller number. Let's work over over C at first. The Miller number equals the number of nodes in this family over T for any sufficiently small a, or close to close to zero. And then Milner shows that this is the same as the degree at the singularity of the gradient. So, have an enrichment for this. Let's see what happens over over an arbitrary field. Again, the characteristic not to, and then our nodes come in different types. The point can have different fields of definitions. So here's our node, and it can have a residue field. That's not the ground field K field L. And then the tangent directions, they might or might not be defined over L with tangent directions. Determine another quadratic extension. And this was square squared of something for a and L. And it won't matter if we change L by a square. This L happens to be always separable as GA 715. Always a separable extension. And for example, we can draw the nodes over our it might be a complex conjugate pair. Which is hard to draw. Maybe we'll just draw the real real nodes over our. We might have the standard X1 squared plus X2 squared or X squared plus Y squared equals zero. And then the tangent directions we can't draw them they have kind of slope. I and negative I. So they're not there. And this is the non-split node. The tangent directions are not defined over the base field. Rational tangent directions. But if we change this to X1 squared minus X2 squared, then we do get our tangent direction. So this is a split node. Rational tangent direction. So let's let's make a type of the node that expresses some of this. Some of this arithmetic. The type of a node, P with the completed local ring being L X1 Xn is defined to be the local degree of the gradient of this this equation here. Explicitly type B will take the this transfer. Which has a nice explicit description with post composition with the with the with the trace from Galois theory, two to the end product of the AI and GW of K. So then we can define the a one Miller number to be the local degree of the gradient again taken in in Morrell's enriched enriched sense. And this is also the sum over the type of P of the nodes in a generic so there's an open set of a nodes P and a generic family in a family in the family above for generic a Jesse Cass and I looked at that in the same paper I was mentioning previously and Sabrina Polly with her dynamic interpretation. Looked at it. As well. So we can do the above example degree. We above we computed the degree of f of z goes to z squared. And so we can, we can take a look at a concrete example of this particular equality. This is one of many in in Orlick's in Orlick's paper, some of which are not enriched at the moment. So we've got a cusp, which looks like, like this, our characteristic. Okay, not be two or three. And the singular point is 00. In our hyper surface. We have a gradient. We have three x squared and minus to why the degree of the gradient. And this is actually the smash product of the map that sends x x squared and y to minus to why. So, we get to take the product. And this goes to three x squared and y goes to minus to why. For similar reasons to the above this is H. And this is minus two. And one of our relations from last time was that you can multiply H by anything you just get an integral multiple of age this is H again. And so this is our a one million number. So we can then watch what happens in the family I'm going to choose a family that looks like something from from algebra. So if we take the family, we had x squared minus y, x cubed minus y squared. So we can look at the family where y squared was actually x cubed and you know plus some x plus some y but if we do this, this is one of our equations above. And then we have this equation where we were when it has a double root it's a it's a classical discriminate. So, when a equals zero will draw this family. So here's the T plane. And at T equals zero, we have our cusp. Other T we have something smooth. So if we draw that equal to zero, then we have singular fibers. This is a picture of this family over here. And we'll draw the same thing over here but a is going to be something some fixed non zero, but a single fibers, when x cubed plus a x plus T as double roots, which happens if and only if the discriminant equals zero and that's minus for a cubed. This is minus 27 v squared a cubed minus 27 v squared equals zero. So there are two. Where this is, this is T, there are two values of T, where there are there are nodes, which is the these was the square root of minus for a cubed over 27. And you can see that this is bifurcates the cusp bifurcates into two nodes. And this was ranked two. The rank of the the Euler number or the that the number number, number, which is so the, which is the number number so we can see that equality between the local degree of the gradient and and those two. That's right there. But now that we've got an equality in GW of K. This also says something about the kinds of nodes that you can have there about their types so we see that it. So, over the finite field with five elements, then one is equal to minus one. So, the cusp can't bifurcate into one non-split and one non-split node rational over finite field with a three mod four elements, then one is not equal to minus one, and we can't bifurcate into two split or two non-split rational nodes. The numbers are used for some some very cool things. And one of them is when, when you have a family of spaces degenerating to a more singular space in the middle, there are formulas for the Euler characteristic of the more smooth space and the difference between the Euler characteristic of the more smooth space and the more singular space in what's called conductor formulas. So, classical Milner lumber appears in conductor formulas. And it's also related to the the Euler characteristic of Milner fiber, Euler characteristic of Milner fiber. The Milner fiber, there is a motivic Milner fiber it's in canada varieties of Denetha and Loser. And these very, very interesting conductor formulas. One could ask about quadratic enrichments and Mark Levine and LaHoller and Srinivas have recent work about quadratic enrichments of such formulas. And they're very subtle and Ron Azuri has generalized this. So they have subtle and lovely results on enrichments and in GWFK. Let's talk about another appearance of the the Ekl form that comes up in a totally different context and I like the fact that it comes up in this other context as well so for that let's talk about the A1 Euler characteristic, and it's related to this work. So, for a smooth projective variety over K so like a smooth compact manifold over again our field K. We had this Euler number last time. Let's get some more notation for the terms we need to talk about that so last time. We had a vector bundle V is relatively oriented and by the data of a line bundle and an isomorphism where X is a line bundle. We have a square with the harm from the determinant of the tangent space let's say X is smooth and the determinant of our bundle. So one one vector bundle that definitely has a relative orientation is the tangent and the cotangent bundles. So, in TX for V. We get that the home space is this trivial bundle. So since the home space TX TX is just the trivial line bundle and algebraic geometry that's often Oh, for for the regular functions. This is a trivial line bundle of rank one. And so that's got a canonical isomorphism to a square. And that giving us an orientation. So we were discussing last time how with an orientation. We could define an Euler number. Hey, last time we used a section with isolated zeros, but we can take that away and we will this time that assumption about the isolated zero so it follows that we made a fine an Euler number of the tangent bundle and that's going to be the Euler characteristic. This was our notation for Euler number. And there are other candidates for the Euler characteristic. Frederick Lees was giving us the dual of smooth projective schemes or coherent objects in his in his in his last talk. And for folks who know how to take an abstract dual and and come up with an element of the endomorphism to the unit so this was also our maps from s not s not. And we have a categorical Euler characteristic and that's the same as a Mark Levine shows so this is also categorical Euler characteristic although we're not going to define it. Right now. And there's a reference to Mark's paper aspects of a number of geometry with quadratic forms in the notes. So with our, with our Euler characteristic definition let's compute a really fun example that's also due to Mark and Mahaler and Srinivas. They compute the Euler characteristic of a hyper surface. And the case of an odd is is, it's just a multiple of a hyperbolic plane so let's restrict it, and even because it's the more fun one. And here's our hyper surface it's cut out by an equation. For F is homogeneous polynomial of some degree. It means a homogeneous degree E polynomials in these variables. Genius degree E. So, define the part of the Ekl form to find B Jack to be the restriction of the form we were looking at above q cross q omega Ekl to K. So q is now that are the associated function is all of the partials of of F. So let's let X actually be smooth. So q is going to be the form associated to the partials X not X one, etc. And so I just sneakily added the hypothesis that this is smooth. This has an isolated zero at the origin. And so we can run that Ekl form. And we'll restrict that Ekl form to some of the degrees q equals zero to N. capital Q sub little q. These are the degree little q plus one times E minus and minus two. They show that the a one Euler characteristic is this fun element of GWK. It is E plus minus E B Jack plus and over to H. In other words modulo some some hokey pokey. This Euler characteristic is being given by that Ekl form of the zero given by the partials of its of its defining equation. Here again H is our hyperbolic element. And so you can plug this into a computer and get something out. Kristen. Yes, that's that's in that's in my paper with raxid. That's the thing with raxid. Thank you. So let's let's let's get something fun out. So the clepsch cubic surface is this picture that's been hanging around here in my notes. There it is it's beautiful. This is due to the, there's a picture credit in the notes. I didn't take this photo. So this is the clepsch X zero X one X two X three such that the sum XI cubed is the sum of the XI cubed. And in P three. Clepsch cubic surface cubic surfaces over an algebraic closure are blow up P two at six points. So it's classical Euler characteristic is is nine. And it's a one Euler characteristic. You stick this into the computer, and you get out. Two H plus minus 10 plus minus six plus minus 21 plus minus 14 plus minus two. And I did not do this by hand. I computed it last night with Macaulay to we can see that at five. There's some nice residues at five. There's something interesting going on. Plus, it's pretty. You see, you see the residue, you see the five and the existing degenerate at five or something. So away from characteristic five. Like this is, there's another form of it in characters by this thing is bad. And there's another form of the surface that uses that that is another equation that would be good at characteristic five. So we should, you know what we should actually plug that into the computer to and see what comes up. Okay, so I want to add another page now beyond this. All right, our second question that we had ended with last time was, let's make an Euler class that will give us more control over this Euler number. In particular, we had an application for the Euler number being independent of the section we computed it with. And it's better to have machinery to to deal with with these classes for that machinery needs co homology. And we have our stable homotopy category or stable whole a one homotopy theory. And this produces a lot of co homology theories on smooth schemes. And some examples to keep in mind are the motivic homology that we've been discussing, our extended, co motivate co homology that that's also come up. Let's say a little more about just a second. K theory permission K theory, and with a co homology theory. Let's, let's get co homology groups. So, one of the, one of the revelations for me as a as a younger mathematician was getting to define co homology groups with spectra so as an advertisement for stable homotopy even if you're you're just learning it. One of the ways to describe even singular co homology once you have your stable homotopy category it doesn't mean need to be a one. We get the homology even singular homology has this formula as pi minus n of harm from from x to H, which the pi minus and pi n is like homes from s to the end, pi minus and we get to put the end on the other side. And this end corresponds to some sort of shift, and it's very useful to allow a twisted shift, this let's say the stable homotopy category. Let's twist our shifts, yes, twist our shifts by vector model so what the vector bundle. And then we'll say that the degree v part, again will be like the rank of the is now the homotopy classes of maps from x, the Tom space of the smash, smash, smash, smash H. So thinking of the Tom space of the as a twisted shift so this came up in Frederick de Glesis talk we could take the over v minus the zero section for some reasons about taking Tom spaces and K theory let's put a dual, dual there this is also a projectivization. If you take a trivial bundle. Then crush the things way at infinity. And this is like a suspension it would be if this was a trivial bundle, but this is a twisted shift of the space x or a twisted suspension of X. So trivial bundles this gets us back to what we had before. And one notation for the trivial bundle of rank n. Here we go, trivial rank and bundle on on X. And then the Tom space. It becomes just the suspension. And we have what we started with. I just don't know. You don't have to write. So then HV equals h n and then the notation above. So in our examples. We we get to see the connection with with child groups that we were discussing in Matthew Maro's talk. So, this is the C n of X. This is the, the motivate cohomology h 2n, X, the N. And some some other indexing would make this h 2n, and X. And this is the child group of co-dimension and cycles, the formal sums of co-dimension and irreducible sub varieties, child group of co-dimension and cycles, modulo rational equivalents. So we have our motivate. Yes, good. Yes, can we access other gradings. These are the geometric gradings. And for this talk, we will not access the other, the other gradings, but we could with a spectrum with different suspensions. So, the next on our list was an extended motivate co-homology. And then the coefficients of the sub varieties are elements of the growth indeed that group. The chauvit group, also called the oriented chau groups. And with the Gersten resolution from Frederick de Glese's talks, we're going to express these as formal sums of co-dimension and sub varieties. Whose coefficients, the growth indeed that group of the rational functions on C, and then they have to vanish under a differential. And modular the image of another one subject to conditions, modular equivalents, and their references in the notes to barge Morel's original article, and to a recent book about Milner bit motives by Bachman Kalma's Faisal Ostpier. We had two more examples to keep in mind and we've already been discussing K theory. But let's let's just look at K naught. And this is the group completion of vector bundles on X, then we have KO permission K theory group completion of vector bundles with a symmetric non degenerate form with asymmetric non degenerate by linear form. So, these, these theories are representable because they're expressed as this harm from X to this, the theory in this stable homotopic category. And for such theories, we can also have homology with supports. So we'll take a closed subscheme of X, and the homology with support in Z, and set of homotopic classes of maps from X. Let's put the support here we have a twist here of X. So we have homotopic classes of maps, which are trivial off of Z a few potion by the compliment, take maps out of this, then the map is trivial away from Z. So we have comology, the comology theories, and we'll have an Euler class for these comology theories. H, co homology theory will make H a ring. And then for our vector bundle with a section but the section could be the zero section section, for example, the zero section on the other class is the the co homology in degree of the dual supported at f equals zero of X. And with respect to H to. Yeah, here's the H. And here's the H. And here's the Euler class of the map, where we take f. And this gives a map to be over the minus zero section. And we can use the map from from S to H to smash with H. In fact, everything that's not in the zeros of F is sent to the base point. So we have support as claimed, and as our element here, our Euler class. Our Euler number, which was the sum of a bunch of local indices. So the Euler number. And to have a push forward and to have a push forward, we need some assumptions on our on our maps. Let's, let's put some some in. So, these assumptions are so that the, the duality theory has something to do with the, with the tangent space or co tangent space. A function is a local complete intersection morphism. If it locally factors, as a closed immersion followed by something smooth, and the closed immersion has to have a nice casual complex. So what it locally factors as Matt P, which is smooth and a closed immersion determined a casual regular sequence. So a regular sequence is when, when you mod out by the first I, and then mod out by that last one. So it's a zero divisor. And, and if the casual is that the higher comology of the casual complex is zero. And the properties we want is that it has a well behaved co tangent complex which is a generalization of the cotangent space. So, this has well behaved co tangent. And the, the few facts that are needed here about the cotangent complex is that for the closed immersion, it's the normal bundle or really it's dual. And for something smooth, it's the, the dual of the relative tangent, tangent bundle. So, we'll call the co tangent complex of F, call it LF. And for one of these regular embeddings, the co tangent complex is the normal bundle of the, this, this closed sub scheme here of this. And it's really the dual like co tangent instead of tangent and co normal instead of normal and degree one. So, the dual of the normal bundle. And for something smooth, it's the relative tangent spaces dual for the scalar differentials mega p over s so the, the fiber wise tangent dual and then the composition of the co tangent bundle is determined by these two. I star LP to L PI to L to Li. And so we have some relation between sero duality and the co tangent space for for these kinds of, for these kinds of maps. And we also have a push forward for these kinds of maps. So let's let P x to S be a proper for the inverse image of compact as compact LCI that definition above. And we have a map the other way So, this called the Becker Gottlieb transfer Mark has a version that's been sort of push forwards are also sort of part of a general theory by lots of the wonderful work Frederick de Glese was was telling us about so I mean there are lots of points of view about what's happening here let me draw you a cartoon but in general if you have X to S in the stable homotopic category you can turn it around and get a backwards map from S to X but no it's not quite to access to a shifted twist. So it's to the Tom spectrum of LP so a cartoon for this is so here we have X to S this isn't the dimensions I'd really like. But if you embed X into a bundle over as trivial as and I mean to get push forwards we should follow Frederick de Glese's description from from yesterday but to give a picture of how this space comes in, you know we embed this kind of up here, and then we do a collapse map of some sort of neighborhood, and the Tom space of this neighborhood. You should do a fiber wise collapse we get the Tom space of the trivial bundle over as we can collapse where over this region will do fiber wise and this is the suspension of us this Tom space is is a twisted shift of X along something like this normal fiber wise normal bundle. So it's a cartoon from say stable homotopy theory and once we have a map, the other direction, we get maps on comology theories to and then I'm feeling a little guilty for this cartoon. We should really do these with the six function formalism that we saw last time and I use purity and the vod skis work. So this was, this was sort of unfair. It's fine. Nice. All right, thanks. Thanks for the for the for the indulgence. Okay, so we have we have a transfer and it should look like something to do with the cotangent complex. With this, with this, this transfer. If our theory is oriented, we're going to be able to push forward our order their class to get an order number and the kinds of orientations we're going to be interested in our kinds where we could potentially sort of untwist untwist here and what is an orientation on a theory, it allows us to take a twisted cohomology group and untwist it. So that's so oriented cohomology theories. So, H is geo oriented. If the twisted cohomology groups are untwisted. Yeah, for and their canonical maps. Giving these isomorphisms. For example, motivate cohomology and K theory, but not extended motivate cohomology and permission K theory. So that's too bad it means it's harder to push forward. But H is SL oriented. If there are canonical isomorphisms between the between two different twists when both their rank and their determinant, as in the determinant is one. If there are canonical isomorphisms between the V and V prime degree twisted cohomologies. If the rank of the equals the rank of the prime and the determinant of the is the determinant of the prime. There's an isomorphism here. So it turns out that this these are the same theories and a canonical way so an SL orientation on our theory gives us something called an SLC orientation, which allows us to make the isomorphism between the determinants, the only an isomorphism up to squares. So for L, X a line bundle. I want to give a hats off to some beautiful papers of N and yep ski on SLC oriented cosmology theories. And for, for this while weaker notion here, we get the extended motivate cohomology and KO are in fact SLC oriented. So under our definition of a relative orientation, it was cooked up to be able to push forward. So this was that the home from the tangent space the terminal tangent space, which is like our determinant of the L, the LP here. Yeah, so let's let's give ourselves a relatively oriented vector bundle DX relatively relatively oriented vector bundle on X PKP smooth proper H, an SLC oriented theory. Then, we have the the co homology in degree the dual is isomorphic to the co homology in the in the tangent space and also for something smooth this LF that we had before is is that so let F be any section of the, for example, the zero section. Then we have inside H, the dual of X with support and F, we had to find above the Euler class of the F, and we can, we have a forget support map to the degree the dual homology of X with no supports. So any two sections F one F two of the they don't have to have isolated zeros or anything so they they're just living in that in that vector space of all sections. So they're connected by families of a ones they're in a vector space by straight lines and this vector space H not a V. This implies that the that the when we forget the support of the H, the effect, what is the forget support map, it's the class of the map that goes from X, X over X minus F equals zero to the Tom space of the X and this was just, you know, F plus smashing with with S, but more or less it's F. And since these are homotopic, we have that when we forget support. It's immediate that the image, the image after we forget the support is equal. So we can call this can just define this common class to be EHV here. So what we've got so far is that under IOTA, we are now independent of section. And then we did work so that this was HLP X. We have P star to H not S. And what is the Euler number it is the image under P star of this. So that mission, the Euler number in HV of the and H not of S is the push forward of E. So then we had a sum of of local indices last time for doing counting. And then we had degrees with the NVF, which was a sum over X and X, where the section was zero of a local degree. And that our local degree was in GW of K. So where we define this local degree with with some comments here. H equals H C twiddle K O over S equals equals spec K. So, and in particular, our second question, which was why is this independent of the choice of section is now is now for free so this agrees with that. And this gives so we now have that and the F is independent, the choice of section and some references are de Glesion cons paper that we talked about last time, and some joint work with Tom Bachman. I think we paid back the IOUs from from Wednesday. So, let's let's use this as promised to to give an arithmetic count of the lines on a smooth cubic surface. And the joint with Jesse Cass. So as mentioned before, a cubic surface is so X and P three X equals F equals zero where this is. The homogeneous will we'll call them W, X, Y, Z of degree three. This is the zero locus of a homogeneous degree three equation and there's this beautiful classical theorem of Salman and Kaylee from 1849. It says any smooth cubic surface over C has exactly 27 lines on it. And here's here's an example I wish I'd left myself a little bit more time because there's a lot of fun geometry here. But I didn't. I hope I have time to do this. So, let's look at the from my cubic surface, a little less beautiful than the collapse but so it goes. This is f of X, Y, Z, W is the sum of XI of XI cube. So we've got three real lines. And the real lines, we could so in homogeneous coordinates on P three, we could take any ST and P one and just do s minus ST minus T it'll cancel out met some of the cubes. And this is a straight line here in P one C, and this is indeed in X. We can do the same trick but with other routes of unity, but they won't be real lines this time. So we have S lambda s t omega t for ST and P one C. We can permute these, the variables, we get to choose which ones are SS and which ones are teas. So the three ways. So this gives a total of three permutations three omegas three lambdas three times three times three equals 27 lines. So there we go for the for the for my cubic surface. So I'm on Kaley's theorem is is true. We'll do the proof in general. Let's view the lines on the cubic surface is the zeros of a section of a vector bundle so the space of all lines will denote gr 13 so keeping with our earlier notation. So this is the gross monion of vector subspaces of C for dimension of w equals to or equivalently the lines and P three P ones the projectivization of w and P three the projectivization of C four. We have the, the tautological bundle. And to remind you what that is, it's not the canonical bundle. It's fiber over a line projectivation of w is w. If we want to see what ones are in the surface we need to know when a cubic polynomial polynomial vanishes. So we can make sin three of a vector bundle just like a vector space. It's, it's fiber over a point is then sin three of the vector space over the point let's take some 3b dual. This is cubic polynomials. And we have one that we're very interested in whether or not it vanishes. So f, the equation determining our cubic surface determines a section sigma f of some three of s dual sigma f at this line projectivization of w is the cubic polynomial restricted to w. And, and then sigma f as a zero, if and only if that polynomial is zero which is if and only if that line is in the surface. Sigma equals zero, if and only if the line P w is an X. And we've now reduced to counting the zeros of a section. We have our Euler number. And it's the sum over the lines. L and X, the zeros of a section of a local degree at L, you're denoting LP of w before of sigma of F. And for a general hyper surface or even complete intersection. The zero locus of a section is always smooth by a result of the bar man of L and in fact, for cubic surfaces, there's so many wonderful things known about them is that so for a smooth cubic surface. That's a generic cubic surface in particular but we even know sort of the geometry. They're no more higher multiplicity roots so for a smooth cubic surface. We have all zeros. Sigma f of multiplicity one. Since it's overseas. If we were using the degree in singular homology the degree from differential topology, we would get it follows it. The number of lines that the patient is preserved so we have that this is just one for all of these in our in our classical context. So we get that the number of lines is the Euler number of sin three of the dual we started calling this V for our vector one. So we can compute that and a V is equal 27 by knowing things about the homology of grass monions or even with the from my example that we just we just did. Because this other number is independent of the section. So we can compute this with the example that that we chose. This happens over C and cubic surfaces over are there's there's also a lovely story. So, there's been a lot of work on this, but some notable work is 19th century worth do dish lovely, and there can be three seven 15 or 27 lines. We separated these into what he called hyperbolic and elliptic lines, and they're hyperbolic and elliptic elements of PGL to. And the words are compatible. Let's let's do the geometry of real lines. It's fun. So, if you take L and X, a real line. We're called the definition of hyperbolic and elliptic elements of PGL to and we, I think we're going to have to skip that. So, but L gives an evolution of the line which in particular is on PGL to the evolution L to L. So, it, there, it turns out there exactly two points. I'll just defend it. So, I have P is the one other point that has the same tangent space to X at that other point. So I have P is defined. We take the tangent space the cubic surface at P there it is it's a plane. We take the line, because it's the tangent space and the line is in the surface. So, the tangent space intersect with the cubic surface X has to have the line and that has to have something else of degree two. We take the line in it, and it's got C for C degree two, because the total degree has to be three P, the points of intersection of C with L, two points. And the other one is where this is sent to on the evolution. This is precisely the one other point whose tangent space. So, a tangent space of P at X is the tangent space of Q at X, and the evolution is switching them. So we swap the two points that have the same, the same tangent space. And then this involution has some fixed points. The fixed points of this involution they might be a C conjugate pair, in which case, the line is elliptic. So it's an involution that has two fixed points so the, and those two fixed points they're either both over R or a complex kind of conjugate pair. So elliptic, it's just called the line is then called elliptic. And if they're the fixed points of I are two real points, we say the line is hyperbolic. I can say this better, we say, elliptic and here we say L is hyperbolic. I cannot resist, even though I don't have time for doing this to tell you what's going on here. So, we're interested in the tangent space to your cubic surface you have a cubic surface in a room, you put your index finger along the line, and you let your palm follow the tangent space. So there it goes it's following like this. If you spin all the way around, you are elliptic. If you wobble, but don't spin all the way around you are hyperbolic and so you can check what they are for the Fermat I left a picture in the notes just that that sort of the pictures explanation is that there are three real lines on the Fermat and every place in the picture where there's a plus. I can draw. So here's the Fermat. We have three real lines for my like so they're all in a plane. And here's someplace above the plane where the cubic surfaces and here's below. So if you follow your hand, you first along, say this line, your first like this with it like this, then we go this, this is are all the real lines are hyperbolic. It's a literal exercise because you are literally moving and I love this joke. Okay, so the theorem is that once we separate these, I'm going to take just just two minutes. So sorry about this. So Segre. I took time in the beginning so go for it. Thank you. Segre could have could have noticed this he has enough tables, but it's not clear that he did. And then there's some lovely papers by a Konec Telemann that are that are really quite recent and finish and Karlamot. And from the spin pin point of view, this is in a paper of the Nenadeti circle. And you can get it from more of an Solomon's open Gromov Whitton invariant point of view. And for any smooth real cubic surface, the number of hyperbolic lines minus the number of elliptic lines is equal is equal to three. And the question that we're now all set up to answer is what about other fields. What about K equals Fp, qp, q, etc. And our answer is that the above proof still works and they went home with the theory. So, the above proof goes through. And here's what we get. Let's give ourselves a line L in X, a cubic surface smooth inside p3. And we were interested in how the tannin space spins around the line in this evolution. So let's look at the fixed points of that in dilution again and make a type. So the type of L it's going to be an element of the growth indeed big group of the field of definition of the line. So D is in KL star or KL star squared so that we can put it in brackets with our generators. So is such that the fixed points of our evolution, those two points where the tangent space pauses for a second, as it spins around the line, the fixed points of the evolution of I are a conjugate pair of points defined. Over KL, a joint square root of D. And there's some other ways to say this in the notes. And the, it turns out that the field of definition of all these KL are separable so our favorite way of writing the transfer as post composing with the sum of the Galois conjugates will give a transfer on GW of KL the GW of K. And here's our theorem. Let K be a field of characteristic, not to let X be a smooth cubic surface and P three, then the sum of all of these types of the lines. And the sum over the lines of a waiting that's associated to the field of definition we take this transfer on GW using the field of definition, and then this type which records the pause points of the tangent space spinning around the line. This always has to be 15 one plus 12 minus one in GW. Okay. So, in the notes you can see the comparison to see over are some fun counts, making a kind of an even or odd statement for a finite field. And the fact that for Quintic three folds there's a very beautiful type due to Sabrina Pauley. And also with some earlier work over are by Finesha and Karlamoff. So that there's a lot of great great geometry to say here I am I am over time now so I will I will leave it at that. Thank you very much. Oh, thank you, Kirsten. Are there any questions. Maybe I can ask. Once you you you see the hyperbolic and elliptic. It's not just really do something with its normal bundle. So I want to know if they really need to define a cycle in the choice group because to define a cycle in choice group it's in kind of needs some condition with different There are no bundle at their intersection so do you have some kind of results. Can you say that again. I'm not I'm not following yet. Normal bundles for deformation. Go ahead. Yeah, I mean, I mean, you know the show it group right. I mean, to define a cycle in a show it group, you need some condition, I mean, of different, and I or different piece, when they're intersection we need some condition of their normal bundle. Do you mean in the product structure on the product structure and show. So, so, can you, can you mean by this result, can you give some cycle in the choice group, I mean, some long trip. So, the, there's a there's an Euler on Euler class, which would put the little local indices at attached to the particular lines on the cubic surface on the gross money and, but maybe that's not what you're asking for. So probably the Euler class with with respect to a particular section for a cubic surface you get a bunch of little points for your lines, and their coefficients would be the local index, and then that's on this cross money in. And then I don't know about classes in child bit of the cubic surface as a way to get the, the local index. So, another interpretation of the type in terms of child bit of the cubic surface. I'm not sure but thank you. Another question. Thank you. Thank you. Yes, yes. So I have one question and one comment. And so maybe first question. So you were, when you were talking about a one milner numbers you also you also mentioned these. Motivate millner fibers right there, which live in cannot. And there's also this this motivic monotomy conjecture about these motivic millner fibers. This is also somehow relatable to your a one degrees or. Would you tell me what the motivic monodrome. So, in very wake terms it says that poles of certain motivic zeta functions correspond to eigenvalues of the monotomy action on some point of the motivic millner fiber. Cool. I don't know mark might have some good comments about that. Maybe. Maybe I might. Okay, unclear. I know, I looked at that a little bit. And this monodrome that comes from you have a finite group acting so roots of unity acting I guess right. Yes, these classes in this sort of refined. You have a K naught bar where you have this that you know limit of the roots of unity acting. And my impression is that if you take the quadratic refinement. All the stuff that corresponds to roots of unity besides plus and minus one becomes hyperbolic because they get paired with each other you have zeta and zeta inverse. And the reason that you have this kind of duality pairing between the makes things automatically hyperbolic. That's a guess, but the interesting you get an interesting quadratic invariant corresponding to the lesson minus one in that story that's that's my feeling but I don't know what that has to do with the conjecture I think it's I'm not sure if there's a quadratic analog of the conjecture. And yes, so I was wondering regarding your proof of 27 lines on a smooth cubic surface. So it looked very similar to me. So so it reminded me very much of a also somehow motivic in this K naught of variety sense proof of I think galkin and chinder like 215 or something. So what they do is, they use the etal Euler characteristic and compute the etal Euler characteristic of the final variety of lines, right, which also somehow very much corresponds to like the these this third symmetric power you were using in the over the space of all cubic surfaces. Pardon. Are you, are you, is this are you taking the variety of all lines on cubic surfaces on one fixed on one fixed cubic surface yes. So they obtain some results so actually they don't even need smoothness. So they can compute the etal Euler characteristic of any cubic surface for this final variety of lines for any cubic surface and then they also somehow diffuse all these classical results like like you did and I think it could be quite interesting for you maybe I mean it seemed quite similar. The lines on the cubic surface would just be seven points. Pardon. It's a field extension of the 27 points over the base field. Yes, it's like it's like the discriminant of if you if you spread it out over the the moduli space, you'd you'd for for every for every point in the moduli space you'd be looking at the field extension where all of the lines were defined. Yes. I was one more time. This should be shinder and galkin 250 I can also just post you the link in the chat. There, I have a comment about that there's a work of. I was going to say that too. Sarah, did you know you know about that. Yes, I did, but you can say I wasn't sure I didn't think I didn't know you knew about it. Go ahead, you tell. Oh, I. Mark, thank you Mark. So, Eva Bayer flukeger and and Sarah can compute the discriminant of that the space of the lines over the moduli space and actually get the express the quadratic form. That's the discriminant of that generically a tall extension and they can do this. Looking at the vile group of E six and things that that says about the form, and that actually it says something different. It gives a quadratic enrichment of a count of lines with this sort of sums of trace forms, and then it does vary with the cubic surface you have the other characteristic. And so we could plug in to their result the other characteristic that we computed with Macaulay to for example at this talk, and then say that the, that other characteristic has to be the sum over all the lines, not of this way here. But just the trace form. And so we don't have invariance of number, it will depend on the cubic surface, but we have a formula for it. Thanks to mark and racks it and Eva Bayer flukeger and Sarah, which just gives something different. So if you combine that in the notes we didn't get to it, we have this condition on hyperbolic and elliptic lines of parity condition over a finite field. And you can combine their results to show that the number of elliptic lines over a finite field that it's always even. Whereas, if you do this alone you just have it if they were all defined over K. So, yeah, it gives something similar. Thanks very much. Thank you. Nice question. Are there any other questions. This, this might be a bit of a stretch but can you think of this formula as some throw some fake gospel in a type thing where you have balancing out curvature, giving you a constant on the right. I don't know. It is gospel in a no curvature, no curve. I'm going to say a little bit more than that what are you, or I was going off the hyperbolic versus elliptic it felt like positive or negative curvature and you had the difference summing up to three over time and maybe here in a more arithmetic, so this is obviously the way it's a similar appearance and apparently more things that's right. Or did you just say you can probably. I misunderstood maybe but I didn't realize you were talking about the local invariance in those terms and that's an interesting thought. Gerson, you have any comments on that. No, I thanks. Any other questions. I have one. You might have said something about this and I thought as I missed it but you computed the other characteristic of the club shirt surface. And then you have that classical one. It's like how this a one, whether characteristic compares that class four more great how to how to those relate to each other. Great. So the rank of the other characteristic equals the classical one of the C points. And if we were over our if we take the signature of the other characteristic. I think I actually did manage to say this but it was a similar thing was that this is related to the degrees because there's some of local degrees and we have this diagram for the degrees you would have had to take that and extrapolate. Yeah, thank you. That that is. Okay, great. Thanks. Yeah, thanks. So any other questions. Um, so, could you comment a bit more on what the equivalence for the oriented child groups kind of looks like. Um, so we had that Gersten or that Ross Schmidt complex. And so we had some homology groups. We've got the sum over that we take X. Co dimension and we're going to wind up with G w of K of X and then we're going to have some boundary maps here. So we have dimension one greater. And this will hit and came on or that negative degrees or the big group. And then it's going to get hit. Okay, no nervous. One K of X. There's a boundary map here. The boundary map it's like the boundary map and no more K theory. So it comes from maps. So K. Discrete value field are ring of integers. K residue field. And we have a map from K Milner bit one of the function field here. It's going to specialize onto the function field here. And we need to make a map. And then we're going to have a B double your K Milner bit zero of the residue field. We can do this. No, nothing. No nurse. Boundary boundary map for K Milner bit. And this is the Morrell's book. And you want to write topology over a field. It's characterized by if we've got a uniformizer and then a bunch of folks in the integers, we can get rid of the uniformizer and reduce all of these into the residue field. So, you know, you put all these together and you get a boundary map. Maybe it would be good to see a relationship between something like rational equivalence. And I feel like you can sort of see that by analogy with maybe K Milner. If anyone else has a more geometric thing that might be helpful to say, please, please take the field. It is a little messy. When I'm when I explain it, it's a little messy. It sort of looks like that. I think that's what it is. So maybe I can add a comment on this. I mean, as I, I see it. This orientated to go to it could be kind of. We may not just need to the cycle of some piece, but we will need some information of the normal bundle of this piece and when they're in that section there. There are normal bundle where have some condition. That's right there's there's a this was sort of a quick sketch there's extra it's a little more complicated you have to try the normal kind of thing, but just quick sketch. So the determinant of some other questions. I'm curious about the questions you're asking at the end of Professor Moro's lecture, and in particular I was wondering if like the understanding of the K theory of Hermitian K theory how that plays into like these finding these arithmetic sort of counts, or is that not really the local indices for Hermitian K theory are and they're they they agree with the local degree as as a bilinear form. So if we took the K theory, Euler class and the Euler number. We're just going to get something in K zero which is Z, and we'll just get the the rank, and we'll get the answer over the C points. So the fact that we have some homology theories that have interesting homology groups associated to them means that you can, you can get counseling in those groups. Yes. So does that kind of answer your question. Oh, yeah. Yeah, thank you. Okay. Some other questions, comments. Okay, a notational question and then a follow up question. Does the C and SLC orientation stand for anything, or is it just, that's a good question. I don't know anybody got me. So, next question. Okay, my next yeah I only ask because it's sometimes hard for me to remember the condition. So you talked about how, like, KO is SLC oriented, are there. Are there homology theories that are maybe not SLC oriented but when you pass to like some higher so like, instead of differing by a line bundle square, if you differ by, and you probably know where this question is coming from. If you differ by some higher power, are there existing homology theories that are oriented in this sense but not in like a less restrictive sense. Yeah, I don't know. I think you can do things with like, you know, co-board ism for different classical groups like, right. What is it like him, some plectic some plectical board ism is some plectically oriented. Yeah, but he's asking specifically about this condition. Right, it doesn't, right, I think the power of the line bundle I'm not sure about that. That I don't know about. But I mean, but note that Mark is pointing out that we have lots of So, we can take M of B string and of the GL maybe it's just M string like the unit the time the time take universal bundle universal bundle on the B strings. Tom spectrum, we would get another orientation notion of orientation and like TMF has a string orientation, we've got GL and SL and, you know, spin, or it's not quite we're not allowed to do string. We can take the Lothnikov tower. We can then, as Mark is saying we can take the Tom spectrum, though the board ism, which is the Tom spectrum on the universal bundle of lots of cool, lots of cool groups, and have cool orientations. Yeah, that's awesome. And then my, my last really naive question with like, so would MGL be GL oriented and MSL be SL oriented or is that just a, yeah, because it's actually a map from MGL so an orientation is also this is. Yeah, so that that's all very consistent thanks. Yeah, great thank you. So any other questions. All right, well, let me call the workshop to a close. Thanks again for everyone for attending. It's been a lot of fun. And thank Kirsten one more time for a lovely pair of lectures. And at the same time, thank Mark for all the organizers.