Ronald van Luijk, The geometry of del Pezzo surfaces

in silence can hear the next speaker way is around by music and the perilous journey of competitors thank you thanks for watching for giving me all these sets news already that’s very useful and indeed I will be talking about the best services in their geometry but the notes that I’ve used for this which are very nice and i highly recommend three that at your book of them are by anthony there are 0 and you can find these on his website and it’s actually called arithmetic of the pets or services i will try not to say too much about arithmetic but just like Tony it’s the field I come from so we’re going to use to look at these things over the feel of rational numbers for into them try to look at the rational points that are on them so actually we use positive characteristic but our goal is usually rational points still sometimes every here and there you’ll see some of my motivation some theorems that i’ll try to sneak into this come from positive characteristic and but just like the last stop most of what i was saying today will be in any characteristic and in fact i will not even assume that that might feel this algebraically closed nor even is perfect so i will just use k any field and and i will want to use Galois automorphism so that means that if i want to do that i should not just look at the algebraic closure but i should look at the separable closure que el que and this separable motor is contained inside k bar which is an authority and even though the last talk was mostly over algebraic closure and we’ll see that some old states some of the results actually over else the separable closure and which we can then use day and when I say variety last speaker said if we talk about herbs is always smooth and projective I will not necessarily do that get some of a variety for me will be just a separated scheme of finite type / k and but mostly I will often you look at the smooth projected ones the nice ones and in fact that’s exactly what I’ll kollam by definition a nice this is the definition of the word nice a nice variety smooth projected and do you match up the intro right and it means that without this word nice a variety might not necessarily be irreducible not even necessarily reduce but if it’s nice then it’s actually false and not only is it both even when you extend 2 l’s very closure it’s still both irreducible and reduced together as the same as it okay it’s not just because of my arithmetic background that I’d like to follow a theorem it’s also something that we will use later and here’s a theory it’s a theorem I lend an issue and it says that if you have an isomorphism between varieties of course if one has a point so does the other because there are some orphic everything is the same thing but so often we’ll look at by rational maps and then you still want to conclude from information on one something about the other and it

says the following namely net and the irrational map of a variety hey right and assume and I wanted no support then we assume that X prime has a smooth 10 point X prime hey this is if I have a variety and I put a que el jabrai between parentheses behind it is a set of K points or the set of horns over at algebra in this case it’s kate’s so I assume that this is not empty so it has a point open and in fact I want to say that I have a smooth port so here is the point and X proper these are still assumptions then X also has a port and the proof the proof is by induction on the dimension of X crime which are called n for now and so here we go let’s prove this suppose well so if n is 0 then it is pretty obvious because if X prime is just a point and you have a rational map directly to find somewhere so there needs to find there so it maps to a point in X so now let’s assume that n is positive so take such a point P and X prime smooth and take the blower and the blow-up was justified well that’s great so I look at the blower of X prime NP this Maps X Prime and then I can compose this with the map to X now because X is proper the valuated criteria properties tells me that this rational map is well defined on an open subset of co dimension at least two and inside here is my exceptional divisor and this exceptional provider tip well this exceptional divisor is isomorphic of P and minus one so it has co dimension it has the dimension n minus 1 so given that the locus where this rational map is not defined has dimension less than this if I restrict this rational map to e it is still defined on an open set because it cannot be contained in the locus where it’s not defined and that means that now I have a rational map from a lower dimensional variety 2x it still has smooth points over K so by induction I’m not and in fact it doesn’t work in the much for 1300 coast really do i need to do need to start by induction for any good one okay exercise ticket works everything around something else okay good any case on you have a smooth curve and mapping to a projective variety it’s going to be well defined everywhere anyway so actually it does actually had to place where he takes properly are you so much to me well if x were not proper I guess this map might not be might be the not defined that a people are Jersey okay maybe the question is is it easy they’re stupid they don’t seem easy comfort zone another exercise I just saw off the other one so now you have this one instead well i guess i guess you can have here’s the answer to the exercise and I could have a service of general

type where the set of set of points contain an online book and I met to itself with that line the moon so okay so this also shows by the way about in order to write it down from this you can the corollary of this is that by rationally equivalent varieties so by rational map back and forth and the maps the rational points on one are the risky tense if and only if there’s a risky dance on the other one given that I just solved two exercises here is now another one so you still have one what we just heard in the last talk was something where and it was clear that this was well it was stated for the case that there was a gamma field over a perfect field I’m not assumed perfect at all but instead i look at not the algebraic load of a perceptible phone so I’ll restate that theorem and namely is given by mighty and it says that if X is a smooth projector geometrically ejectives and geometrical rational then x is x rational the fastest service or rational component or both some of the better services are also common bond and this does not need anything and we have it soon and I won’t prove this but what I’ll do is sort of L better serves out of things that I want to focus on now so let me tell you again what they are even though we just seeing the definition already so conditioned and I’m also assuming just like the last figure that we already know that you have service a nice surface it’s a smooth projective geometrically integrals that divisors are linear combinations of curves on it and curves you can intersect on a service and if they intersect transversally you can just count the number of points that gives you an intersection number and if you replace one of them by letting the equivalent divisor then the intersection number doesn’t change and that means that you can define an intersection pair on the thakkar group by placing a divisor by a difference of two other revisers that avoids some components that you want to intersect with you can even define self intersection numbers just like we’ve already seen the self intersection of exceptional curve that you get from a blow up for instance as self intersection minus one and here we’re going to define the following the better surface over okay is a nice day sir it has all those things that we want the service we’re at the canonical sheet and well the anti canonical sheet and we can find a degree of such a dull pets or service could be the self intersection of that phenomenal poll divided so sometimes I write sheets and sometimes when I right into this intersection pairing I will

usually think about divisor classes so this she’s here I will usually identify well if you think about the pic I group we’re on a nice surface smooth projective everything is nice so I will usually think about this pacar group as the magnetizer class group and i will write then the class of economical divisor as KX and the degree is the self intersection of Seconal for the back and what is what that ample mean well we’d already seen it in the previous talk but let’s restate it that means if we look at this drink so we look at the 80 x with now but instead when we’re defining the cadaver dimension we were looking at powers tens of powers of the canonical sheath now we’re looking at the tensor power of minus economical sheet because if you look at you have general search the general type and you have lots and lots of differential forms regular differential forms and now we’ll have very few it’s the opposite but if you take the inverse third in the negative of the canonical sheet and we look at minus n then you get lots of sections namely enough for X to give enough to give it embedding of X sine projective space and if you’ve never seen this before and I will now make us more explicit with some examples oh if you have any questions you should also always ask or if I’m going too slow or too fast doesn’t clear is really silent he generated yes I was mentioned in the previous to that like the other one it was simple example see it much easier is it easier here it was nice key simple if you use easily or something young after all this yeah you see something with sure otherwise I think you need to do so in general not even true average of diversion but be sure you do is smoke for EM probably a wearable device it is the primary it’s like people open from oh icic grip okay so yes this is an otherwise if you want to say if you wanted it you’re saying if you’re defining what happened so exactly or is there something i should say example one hour cubic surfaces so i’m taking my ex and III given by one cubic equation in 20 degree then and if you have any complete intersection which a hypersurface certainly is you can give a formula for the canonical sheet the canonical sheet on it is going to be isomorphic with twist of the structure shape namely how much minus n minus 1 where n is equal to three so i guess in this example I’m already doing this slightly more general i’m looking now at inside a TN and i have a complete intersection given by hypersurfaces of degrees degrees d1 up to d.r then economical sheep will be as more 20 x minus n minus 1 for this n plus some of the degrees which in our example so now this poorly sign is back to the example

it’s going to be minus 3 minus 1 is minus 4 plus 3 this is always of minus 1 so minus this is going to be less noble 201 which is the Sheep generated by exactly the coordinates of P 3 and those coordinates give you exactly the embedding into t3 and that means that this she just couldn’t and economical she does exactly give you the embedding that you started with another way said I’m not going to say it the canonical divisor class is a full bag of a hyperplane section 22 your surface so it isn’t the amples it is this and exactly and you also see why here well this is actually for foremost degrees this is what’s going to happen right here you’re taking all these multiples at once but you can also look at the match the projective space that you get from taking the projective space on just one term here in this direct Sun which is what we did here we just looked at the if you just look at the global sections of 0 X to the minus 100 x our own right reminds i’m not home back to the minus more that’s already enough and in fact for these defensive services as long as the degree is not too small it will turn out and we’ll see that later that already just the global sections of the anti kannamma kashif are enough to give an embedding and if that’s the case then you understand why we call this the degree because in general if you have the surface inside a projector space what are the degree of that surface well it’s the self intersection of the hyperplane section because the hyperplane section if you intersect up with another hyperplane you get to the linear space at the co dimension to it intersects your service and a final number of points namely exactly two degree so that self intersection if it is corresponding to that model as you get from the end economical divisor and thus corresponding to the hyperplane planet class it is the degree as you already thought about it but for some cases you actually need a higher multiple try to get another degree of son of a be different and so let’s go to another example namely let’s go to the intersection of two more tricks King before and now I can use the fact that I’ve already written down what it should be for completely sections here so here oh man I will be 0 minus four minus one plus two quadrants of two times degree two plus two plus two this is again 0 minus one and so we can do exactly the same thing and my XM minus one is ample so intersection two quadrants are also an example of course the degree here didn’t write it down did we fear is forward at the great here and we’ll see later in the week Oh or at least the bars part of a talk about these things so I will not say much more about them and instead give you even more example mainly examples of those cases that I mentioned where you do need to look at a higher multiple all day and economical divisor class and so I can also look in weighted projector on 12 given by also passionate is 0 where the 30 ad is for but now this is

the way two degrees i’m looking at a prediction space with say for these exploits MW and for example an equation like this where this has agreed for and how do you compute the canonical denied i do assume it’s smooth maybe I didn’t say that for the other examples but I’m assuming that i’m looking at smooth but then Cody stick to your book is a characteristic to this would not be smooth that’s true so then you take and apparently my example was not in characteristic too so now it is also possible to do in characteristic too and how do you compute the canonical class here well oh my god x it’s exactly the same thing as you do it for the other case for in the normal projective space except that you have to understand what exactly the same means there was a minus n minus 1 there in the beginning and that’s just minus the sum of all the weights because pn regular p n has n plus 1 coordinates so that minus n minus 1 which is minus the sum of all the other number for us here you get minus the sum of the weights love the sum of the degrees of which your economic a complaint intersection so that means that in this case it’s full of minus one minus one minus two minus 5 plus 4 again is oh man okay yet another example and these examples are here the degree if you a little bit more careful in 3 is what Peter degree is not exactly the self intersection of a hyperplane section but because you’re exactly a factor 2 off because the product of the waist is now something that comes into play and in fact the degree here too so these are the best service degree too and as you want del peso services degree one then you have to look at if you take a surface inside p 1 1 2 3 of degree 6 then as smooth then it’s the peasant surface and the degree as the better surface is a 3 1 and the factor that the one and the six are off by is exactly in the product of the ways and due to time constraints and warm up and go into that example right now either and but these are just examples in fact these are very typical examples so every episo– certain degree one two three or four can be given in one of these ways that I just wrote down and that’s something that I want to go into now so Ramon rock on services the following us that and if we have divided on our surface x + H 0 so the division of the nine seconds of d is 81 d plus the HT of the is equal to underneath here because I want you to say something about things happen can t minus the canonical right plus or a characteristic of structure sheet and now if so this guy here is the

same as h0 by siyasi k minus d and this here usually you don’t know much about but there’s something called kodaira vanishing which says that this 0 if he is ample and anybody who knows a diver vanishing should say but that’s only in characteristic 0 and the nice thing is that the canary vanishing is also true and positive characteristics for rational surfaces now we don’t know yet that these are irrational but they are because we’ve seen in the previous talk custom mowers criterion for rationality and because if we apply this for d equal to minus the canonical divisor and this will need a 0 and but also the minus the canonical divisor is ample so it doesn’t have any global sections and two times it is also ample so it’s also waiting over so the canonical one doesn’t have any blown sections so it is a sublease from that criterion we immediately see that that these services are need rational services so kodaira vanishing for Apple devices is still ok so it is zero and that means it is implied and this here by the way we also know that this is one and that implies that we know exactly that the number of global sections of minus M times the canonical divisor and greater than zero is what if you just work this out and just use t equal to minus M times economic advisor to use everything that I just told you we know this is 1 plus a half times and de 10 m and n / fun where D was the degree of X as the pastorship that degree I’m so we’re using that position and it means that now that you know the dimensions of global sections you can play a game that you may have seen for elliptic curves before where if you just know you have a curve of Jesus one with a point you can look at the rim on rocks places associated to multiples of that point to get sections that you can use for an embedding and at some point you have too many so there’s a relation also now you can do that exactly as well so for example for d equals 3 what do we get that well for N equals 1 we get a mac for N equals 1 and D equals 3 1 x 2 divided by 2 3 we had four so we get a map the p3 so you get a map for x 2 t 33 and how does it now you have to do some more work to see that it’s an embedding but after you do that work and yeah but it’s not bad and now you want to know what is the three and surface inside there well you can look at N equals 32 and look at that the space of qubits and the space of qubits we find that the h0 of minus 3 is well so we plug in N equals 3 3 times 4 is 12 / 26 times 93 so this is 119 but the space of cubix expected cubic polynomials in four variables its twenty dimensional so mainly three plus three and so we have one we have 20 of monomials of the green three and four variables but the space should be 19 dimensional so it has to be a relation among them so indeed we get a cubic relation forex inside p 3 so we get a cubic inside p 3 now there’s another exercise do the same thing for g equals four you will see that you can embed them as the intersection of two

quadrants in p4 i’m bluffing a little bit there because the fact they can embed them doesn’t follow from this it doesn’t show you have to show with a lot of with a bit of work that actually is and economical divisor is in fact very important so they do get an inventor and four degree to you can also do the same thing so now for people to look at any just one and you find that too you get three so your match 22 that’s exactly what we have we have the W square it is equal to something some vortec in x y&z it’s a double cover of each and indeed if you take so maybe that’s why and if you look at M is equal to 2 then you get a map from X to a higher dimensional projective space of course if you already have coordinates or functions x y&z in the Riemann Rock space for m equals 1 you have their squid their polynomials of degree 2 so x squared and X Y Z Y where Y Z in this square you have all those they already have a six dimensional subspace of your raymond rock space but for N equals 2 36 to 37 so we have one too few so you have an extra coordinates an extra function W and now he knows I’m y equals four they do all the work that you don’t want to see me do again you see that there is a relation exactly a 34 waited at the great for in here for this must be 6 and in fact because of these polynomials quadratic polynomials and x y&z X Factor’s through the weighted projective space where this embeds into the p6 to Nico and then and so that’s where you get your embedding necessary to it is way too bass drum a 4-degree one you can try to do this game is well okay this is how to power double power and it yes however yeah so it will be separable Iggy get you get a smooth and it has to be you do get an embedding it’s still smooth so it is a separable double cover but the ramification mulpus is it weird because the revocation locus is a double Connor instead the usual for tech so outside characteristic to miss double cover is double combo p 2 ramified over smooth 40 other characteristics still a smooth double coverage separable double cover of e2 but the ramification bogus is a double com can be a double clinic attendees to work no I think it’s always you know it’s always because the thing is it’s given by W squared plus h w is equal to G where h is something of the green 2 and G is something like degree for in XY and z welcome to have a shoot for 0 you cannot equal to 0 th was not zero but the ramification locus is given by kingdom oh yeah and discriminant yeah here comes the discrete also then well you see if h is equal to 2 a CIA agent but I meant to say and it made equal to 0 then this is not a smooth surface so there isn’t happen where each loves it everywhere demon over there the

singularity’s the other the derivative with respect to W is zero everywhere am I in confused no excuse I think you can choose your microscope was no I don’t think so but let me finish this and isolation otherwise i will get confused but the discriminant locus is given by the ramification love is just given by the discriminant so that’s a squared and enough I’m going to be self a family plot it was an orgy yeah oh but for a zero so it’s just a squared so when you get the double partners but so people who H equals 0 I’m inseparable Kosovo but alright Justin yeah but service is humidity it’s not irreducible no that’s not right no now I’m talking oxygen why it should allow the Queen your shakes it’s just it’s just like an ellipse in her 50 y squared equals f of X that’s going to be in characteristic team that’s singular each point in the point where y is equal to 0 well tell you is equal to your hair W yes but wait your partner told Jean but the total x1 let’s do this after work yeah we’ll have a foot breakfast so this is one way to look at these things another way to look at these things is something that many of you have already seen before for his cubic serves the cubic services one would pay his team about a cubic 7 p-3 is that over Ellsbury closure it’s isomorphic to p2 blown up in six points and that is exactly also true for these depressive services and I’m going far too slow I’m now for briefly going to assume that K is in fact separably close and i will give you two theorems that are the analog of what we’ve seen in the hour before and that states that actually things are even a little bit nicer than they were shown an hour ago where it was over L very close to you but you can already do some things over separably closed fields namely and this is the result bike hoops and any sense that you by rational morphism I’m services smooth projector spirit and then already over separable closer it factors as just a budget baller and so then our why we’re x hi I is low in some point and that means that in every step we were one point and the exceptional divisor you get above it is something that has self intersection minus 1 and intersects the canonical divisor with multiplicity minus one except that it might change if you go up one more because something that is useful to know it is if I have and so this is an aside i have x going to y and this is just one blow up so x

is they blow up and what e why so here one here my exceptional heard and then e a self intersection minus one but maybe in why i have curve and a curve that actually he is off and in fact maybe he has multiplicity in on that curse then if I cold hi I can look at and then by upper stomach see so I’d look at the pool back of the curve and X and that pull back that is the strict transform plus a bunch of copies of their exceptional divisor and that is how many Amy exactly m and now you use the fact that other pool of X of this blower the intersection numbers don’t change you find that the strip transform now this is just a matter of writing down now you know that this is equal to minus one and you also know that is another thing that he used this trick transform what’s going through that port M times so after the blow it up that corresponds to exactly any times there is intersects that exceptional curve so intersectional where strict transform with E is exactly m and if you use those two things together with the fact that on the four-way intersection numbers don’t change you find that the self intersection of the strict transform is the same as the stilt intersection of see downstairs except with M Squared taken off here somewhere okay another proposition that was already stated last hour Oh x 15 minutes right then I already used should already stopped oh I should stop in two minutes another proposition says that the minimal so when you’re looking at rational services these minimal models are not unique and indeed what do we have the minimal but this is now minimal I should say k minimal so you cannot blow down anything anymore / k so you cannot blow down any here you see it a broke down so what is it this statement says that to blow ups a unique are blow ups in points that are defined over the separable closure you don’t need to go to an extension you don’t need an inseparable extension to the fine points that you need to blow up and that means that if you blow things down that can always always be done in the acceptable points and you can take a bunch together to form your own borders if you need the key minimal smooth projected rational surfaces I’m still in the case that we’re over separately close field are just Picchu and the usable services and you get by taking owe you one plus some twist of it and and what we’ll see so that means that if you have a better

service over the ground field well now if you want to understand what it looks like over separable closure of reciprocal closure you can what will stop down until you get something minimum then you get one of these and it turns out oh here i should say for n is equal zero or and grade or equal to n 2 but these surfaces contain a curve self intersection minus n and if you blow up more stuff so if you start with a dell penser servicing rose downtown well in the end you end up with this curve that self intersection minus in before and an even lower self intersection so you have curves of self intersection less than minus 1 there’s a lemon out of state after the break that that cannot happen so this is actually not a minimal model of any defensive surface or n greater evil tattoos perennials one this is just P 1 cross P 1 so that could be one of the mineral models that you end up with if she started with a better service and our acceptable flow you start novelties go down and the other one is this and what follows from that out state after you