 This algebraic geometry video will continue the discussion of blowing up from the previous video So the first example we want to do is what happens if we blow up a point of The real plane R2. What do we get as a topological space? So we draw R2 And we're going to blow up this point here So what this means is that R2 has coordinates x comma y and we're going to introduce new coordinates s comma t for the projective line We're going to take the set of all points x y s colon t such that x t equals y s so this will be a subset of R2 times P1 and what we want to know is what does this look like as a topological space? So each None zero point of the real plane corresponds to unique point of this blow-up as we saw last lecture and Zero point of the real plane gets blown up to a copy of P1 over the reals, which is just s1 So somehow we're changing this point to a circle So you can think of this as Having a point for each possible direction through the origin here Now this means that we actually have a map from the blow-up of R2 at a point to P1 of R Because we can just map the point x y s colon t to s colon t and P1 of R is just a circle So we can ask what the inverse image of any point looks like so the inverse image of a point s colon t in P1 is just a copy of the real numbers as you can see fairly easily Roughly speaking if this red line indicates a point of P1 of R And the points in its inverse image consist of all the points on the plane together with a Point above the origin corresponding to this direction So if we write e for the blow-up then we've got a map from e to s1 such the fibers Are all the real copy of the real numbers So what what does that mean? Well the most obvious guess is it might be a cylinder So a cylinder has a map to s1 and the fibers are all real but this is wrong because there's actually some twisting involved In fact you can see that this blow-up is in fact non orientable for example if we take a small Point on Blow-up so suppose we take a point here then we can take a sort of small Coordinate base for it and if we pull this coordinate base through the origin what do we get? Well We get the sort of purple direction still continues going off in the same direction But the blue arrow sort of gets reversed as we go through the origin So we can see as we go from here to here. We've kind of reversed orientation so this must actually be non orientable and The non orientable surface that maps onto s1 and has fibers are is just homeomorphic to moebius band Um, that's an open moebius band. So you don't put a boundary on it. So it's non compact Um And conversely if you take a moebius band looking something like this um and contract Take a midline of it and if you can track this midline to a point what you get is just an open disk Um, so this is a Maybe a little bit surprising because we started with an orientable surface Blowing up a point actually turns it into a non orientable surface the next example Is let's try and construct a map from p1 times p1 to p2 And you can think of p1 as being more or less a line With coordinate x this p1 might be a line with coordinate y and we want to map this to the point x y Well, obviously you can't actually do that because x is not defined at the infinite point of p1. So what's the best we can do? So this doesn't quite work. So let's try again Um, so let's take a point x naught colon x1 in here y naught colon y1 in here and map this to the point Um, let's map it to x naught y naught um x1 y naught sorry x1 y naught x naught y1 then um You might think this gives a map from p1 times p1 to p2 for instance if x naught and y naught are both One then this is the map saying x1 y1 to the point x1 y1 of p2 as before So so this seems to be a correct version of this map. However, this is not defined um at x naught Um If x naught equals y naught equals naught in other words, so if we take the point naught comma one And times naught colon one in p1 times p1 this map isn't quite defined. So, um, this map here is defined everywhere except at one point And the point of this uh, well, we're discussing blow-ups the point is you can make this defined at that point by blowing up p1 times p1 So we're going to blow up p1 times p1 at this bad point naught one times naught one and see what happens um, well Um, we can just cover it with a copy of with a copy of the affine Plane given by the points where x1 equals one y1 equals one So this this affine plane has coordinates in x naught y naught um, and if we blow this up At the point x naught equals y naught equals naught Then we find that this map from this blown up plane doesn't actually map to p2 um, for instance, um, you can blow it up by putting x naught equals t y naught um, then you find The point x naught y naught x1 y naught x naught y1 becomes, um t y naught squared um x1 y naught ty naught y1 And we can take out the factor of y naught and we get t y naught x1 um ty1 And this is now a perfectly good point of the projective plane because x1 is just equal to one so it's not causing problems So in other words, we get a map from the blow up of of p1 times p1 to p2 So what we've done is we started with a rational map here So it's a rational map It's it's not a regular map because there's one point where it's not defined And we found that if we blow up one point Of p1 times p1 we now get a regular map From this blow up to p1 So we've kind of turned a rational map into a regular map by blowing up a point Next we can ask is this an isomorphism And the answer is no The problem is there are two points of p2 that are the images of entire line So if we look at the points naught naught one in p2 this is the image of um an entire line in p1 times p1 um You can see the line is um just the um Set of points um where y naught Is equal to naught in p1 times p1 and similarly naught naught Sorry naught one naught In p2 is the image of the line x naught equals naught Um, and you can check that actually what is going on is The map from this blow up to p2 Is the same as the map You get if you blow up these two points of p2 So what we find is the blow up of p1 times p1 at one point Is equal to the blow up of p2 at two points Obviously you can ask have we used one extra point is the blow up of p1 times p1 at no points Equal to blow up of p2 at one point and the answer is no it isn't you have to kind of if you want to get from p1 to Times p1 to p2 you first will have to blow up one point And then you have to blow down two lines. You remember blowing up a point turns it into a line More generally there are several other things you can do with blow ups so we can Blowing up we can blow up We can blow up a point And the geometric meaning Believe active space of you can think of this as something like the tangent space of p so If you've got a point p we can look at the tangent space Well, we haven't defined tangent spaces yet, but whatever and that's a vector space You can take the corresponding projective space. We're kind of replacing the point by this entire projective space Well last lecture we saw you could also blow up along a line And more generally we can blow up along a sub variety um x The geometric meaning of this is that each p in x is replaced by the projective space of the normal um The normal space At p along x in other words informally you think of taking the Vector space of all tangent vectors that are orthogonal to the sub variety x that gives you a vector space and we take the corresponding geometric corresponding projective space Well, that doesn't actually quite make sense because in algebraic geometry the concept of things being orthogonal to each other Doesn't really make sense. You need a metric for that um But there's a way around that instead of taking the vectors orthogonal to x you can take the tangent space Of the whole variety of x and quotient it out by the tangent space of x or and so on but but informally You think of this as being a projective space of a normal bundle I mean instead of a sub variety we can actually blow up along an ideal And so any sub variety is defined by an ideal um, and in fact You can blow up along more general ideals. We'll see an example of this in a moment More generally still we can blow up along a quasi-coherent sheaf of graded algebras That's a bit of a mouthful. Um, what on earth does this mean? Well, that's easy enough If you've got a graded algebra Can construct a projective variety from it Now suppose you gave a graded algebra for every point of the variety And you replaced every point by the corresponding projective space of that graded algebra That would be a blow up So the question is how do you assign? A graded algebra to every point of the variety? Well, obviously it has to vary in a nice way And the answer turns out to be something called a quasi-coherent sheaf of graded algebras So this rather Intimidating looking mouthful is just a rather complicated way of saying you assign a graded algebra to every point of the variety in a nice way so This corresponds to some sort of Cons you you sort of construct the projective space of a graded algebra at each point So there's one very simple example of this. Suppose we our entire variety is just a point Then a quasi-coherent sheaf of graded algebras is just a graded algebra. It might be something like say k x naught up to x n And then blowing up a point along this graded sheaf of coherent algebras just turns out to be the usual construction of projective space of dimension n and The general form of blowing up is a sort of relative version of this construction Where you do something like this not just at one point of a variety, but at every point of a variety So Hiranaka In a rather famous paper used repeated blowing ups along sub varieties in order to show You could resolve singularities if you've got a variety with singularities and by carefully Blowing up repeatedly along sub varieties. You can obtain a non singular variety in high dimensional space This proof only works in characteristic zero And it's one of the biggest open problems in algebraic geometry to try and generalize this to varieties of non zero characteristic Okay, I said I would give an example of blowing up along an ideal um Or sheaf of ideals never mind. Um, so let's just look at the plane a two with coordinate ring k x y now um blowing up a point Kind of is going to correspond to blowing up along the ideal x y So so here if we take the point zero zero in a two Then it corresponds to this ideal And blowing up along a more general ideal is a sort of generalization of this construction So suppose you've got an ideal with generators g one up to g n Where g i is a polynomial in k x y What i'm going to do Is i'm going to take a squared times p n minus one And i'm going to take the set of all points x Um y g one up to g two up to g n um Now this is going to be defined if x y is not in the sub variety Generated by all these g i's so this is the sub variety of the g i's Because if it's not in the sub variety then at least one of these must be non zero so this is well defined So we take the image of a two minus the sub variety In a squared times p to the n minus one and then take the closure And this will be a sort of blowing up um Corresponding to the ideal um, it's You see the map from a two to this bloke is defined everywhere except on the sub variety and on the sub variety It does something rather complicated and mysterious. In fact, it's actually quite difficult to understand what Hero narka didn't use bloke's long general ideals He just used blow-ups along non singular sub varieties, which are considerably easier to understand um as an example of this suppose we take the ideal i to be Generated by x squared and y squared then um What we do is we just map x y in a two To the point x y x squared y squared in a two times p one So what on earth does this look like? um, well If we call these variables x y s t in a squared times p one then um p p one is covered by two copies of the affine Line, so let's look at one of these. Let's take s one equal one um, well we notice that um x squared t is equal to y squared times s Because s is sort of equal to x squared and t is equal to y squared at least when x and y are not both zero So if we take s equals one we get x squared t Is equal to y squared And this actually has a whole bunch of singularities because it's just the Whitney umbrella all over again, and you notice it as singularities Along the line x equals y equals zero So to be a bit careful about blowing up we've been going on about how blowing up can get rid of singularities If you're a little bit careless about what you blow up you can actually introduce singularities So we've started with a nice non singular variety a squared We've blown off along some slightly strange non reduced ideal and we've ended up with a with a variety that has a singularity And so blowing up along more complicated ideals is a slightly mysterious operation. It's it's very powerful in fact Hieronarchist theorem implies that in characteristic zero you can resolve any singularity in one blow-up By blowing up along some complicated ideal Unfortunately as we've seen these More complicated ideals can actually make things a lot worse if you're not careful. So it's it's hard to tell what's going on Okay, the next lecture will be on another sort of By rational map called the a tier flop