 So this lecture is part of an online algebraic geometry course on schemes and will be about abstract and projective varieties. So we want to compare these two classes of varieties. So first of all, let's just recall what an abstract variety is. This is a scheme of finite type over the spectrum of some field k. And we wanted to be separated and irreducible and reduced. So reduced just means we're not allowing no potents separated avoids these rather funny points and irreducible means it's not the union of two other varieties. On the other hand, projective ones just means a closed irreducible reduced sub scheme of projective space over the field. So these are sort of pretty much classical projective varieties. And we say a variety is complete. So if this variety is complete, it just means the map from V to spectrum of K is proper. And we've seen earlier that projective implies proper. So all projective varieties are complete. Abstract varieties were introduced by Andre Vey in order to construct Jacobians of curves over fields of finite characteristic. He needed this for his proof of the Riemann hypothesis over finite fields. However, afterwards Chao showed that all the Jacobian varieties that they needed were in fact projective varieties. And for a long time it was an open problem to know if there were any abstract varieties that weren't at least quasi projective. So first of all, we can ask is a general abstract variety contained in the complete one. This was answered by Nagata. She showed that any abstract variety is an open subset of a complete one. So this can be thought of as a sort of compactification. So for projective or quite or for ordinary affine or quasi projective varieties, this is completely trivial because you can just complete affine space to projective space. So Nagata showed that all abstract varieties are just you take a complete one and then you take an open subset. So the question is what are the complete ones? So we can ask are all complete varieties projective? And I'll just summarize the answers to this. So in dimension zero or one, the answer is yes. In dimension two, yes if non-singular. However, there are examples of singular ones that are complete but not projective. In dimension three, the answer is no. And a little bit later, I'll give an example of one of these found by Hironaka. Chao has a theorem that shows that complete varieties are not too far from being projective. More precisely, if C is complete, we can find a projective variety P mapping to C, which is an isomorphism on an open subset. I should say maybe better say dense open subset because obviously an isomorphism on an empty open subset. So complete varieties aren't too far from being projective. In fact, quite a lot of theorems about complete varieties are proved by using Chao's theorem. We can prove it for projective varieties by calculation and then just deduce it for the complete varieties C. It's actually rather hard to find examples of abstract complete varieties that aren't projective. Almost all naturally occurring examples that people come across seem to be projective. An example of this was vase Jacobians, which weren't obviously projective but turned out to be. So I want to present an example due to Hironaka of a complete but non-projective variety. This is going to be in dimension three. And so let's suppose that V has dimension three and there's some sort of, I don't know, can be any complete variety. I don't care. Let's choose two curves C1, C2, intersecting in a point. So I'm going to draw all of these curves to be blue and the other one to be red or pink or whatever. They're going to be called, so C2 there and C1 there. We'll call the point where they intersect P and let's blow up C1. So what that means is I'm blowing up along C1 and what that blow up does is it sort of replaces every point of the curve by the sort of projective space of its normal bundle in some sense. The result of this is to replace each point of C1 by a projective line. So if we blow up C1, it sort of ends up with looking a bit like this. See, we can think of that there's a lot of lines there and C2 still looks like this. So C1 is now being blown up and C2 is still looking like that. And the blow up of C2 consists of a line here and called the exceptional curve and the, and together with a sort of copy of C2. And next what we do is we blow up C2. Well, the strict transform of C2, which means we blow it up without blowing up this line here. So what we end up with is a curve C1 looking a bit like this. And if we blow up C2, this again involves replacing points of C2 by copies of projective space. So, so all these lines here are sort of copies of P projective space P1 and all these lines here are copies of P1. And now what we do is we consider a line L1 here. So let's take a line here and call this line L1. So there it is again, here's L1. And we do the same thing here. Let's take one of these P1s and call this L2. And now the key point is that if this is projective, well, if we embedded into projective space, we can define the degree of any curve, which is the number of intersection points of that curve with a hyperplane section countered with multiplicity. And the key point is that the degree of L1 is greater than the degree of L2. And the point that the reason for this is that the degree doesn't vary as you sort of vary the curve in an algebraic family. Now, as we vary the curve L1, we sort of mostly get these lines. But at this point here, something a little bit funny happens because the algebraic family containing L1 kind of ends up being this curve here where it's a copy of where it breaks up into two curves. One is the sort of exceptional curve and one is one of these curves here. Now, this curve here is algebraically equivalent to L2. So L1 is algebraically equivalent to this green curve plus a copy of L2. So in fact, the degree of L1 is equal to the degree of L2 plus the degree of this funny exceptional curve E, where E is this curve here. Well, so far, nothing unusual has happened and we can carry out this construction in starting with a projective variety. And if we blow up twice, we will still have a projective variety. So now Hieronaka did something rather clever. What he did was he took two curves C1 and C2 intersecting in two points. So here they're intersecting in two points. And now what we do is we blow up C1 and C2. However, the cunning thing is at P2, we blow up P1 first. And at P1, we blow up, sorry, not P1, we blow up C1 first and P2, we blow up along C2 first and then C1. Well, first sight, this doesn't make sense because you either blow up along C1 first or you blow up along C2 first. And how can you sort of do both? Well, what we do is we really have to glue. What we do is we remove P1 and then blow up C1 first. Or we can remove P2 and blow up C1 first. Now we take these two things and glue along the manifold where we remove P1 and P2 and blow up C1 and C2. Now if we've removed both P1 and P2, it doesn't make any difference what order we blow up C1 and C2 are because C1 and C2 don't intersect. So here we've got two manifolds with an open subset in common. We can just glue this manifold to this manifold along this manifold. And that's what is meant by this. At P2, we blow up along C1 first and at P1, we blow up along C2 first. You really can make sense of it by blowing up two, by gluing two manifolds together. And so now let's look at what happens. If we blow up along C1, we remember we get this line L1. And if we blow up along C2, we get a line L2. And now we have the following two properties. If this is projective, we find the following properties. First of all, we see the degree of L1 is greater than the degree of L2. And secondly, we find the degree of L2 is greater than the degree of L1. And we find the degree of L1 is greater than the degree of L2 if we blow up along P1 first. If we blow up along L1 first. And we find this if we blow up along L2 first. So we're blowing up L1 first at the... Sorry, I get muddled up about whether this is point P1 or P2. So I'll leave this as an exercise. Well, obviously these two just give a contradiction. So the variety is not projective. And because you can't define the degree of a curve in it in a consistent way. So this example of Hieronokas can be used to produce lots of other weird examples by modifying it slightly. For instance, you can find examples of algebraic spaces that aren't schemed. And break spaces are a sort of slight generalization of scheme introduced by Artin. Okay, that'll be all on proper and separated morphisms. The next topic we want to cover will be sheaves of modules over a scheme.