 This algebraic geometry lecture will mostly be about grass manians. So in the last lecture we looked at the grass manian g2,2, which can be thought of as two-dimensional subspaces of a four-dimensional vector space or alternatively lines in space. Now we're going to say a little bit about more general grass manians gmn, which are injected maps from km to km plus n. And much of the theory of grass manians gmn looks like the theory of the grass manians g2,2, only there's more bookkeeping going on. In particular, we can show that the grass manian gmn is contained in a projective space of dimension m plus n choose m minus 1. So for g2,2 this would be four choose 2 minus 1 which is p5. So how do we do this? Well the details, suppose given some m-dimensional subspace of the vector space km plus n. What we do is we pick m vectors spanning it, which gives a matrix of size m times m plus n. So we might have a vector with points a1 up to a m plus n, b1 up to bm plus n and so on, where each row form the coordinates of one of these points. And now we pick any m columns. So here we have m rows. We might pick m columns and look at the determinant of these columns. So this is just an m by m, the determinant of an m by n minor of this large matrix. So you remember last time for g2,2 we just had two rows and we just picked two columns and gave that the determinant. So this gives us m plus n choose m numbers because this is the number of ways of picking m columns from this. And let's call these numbers p i1 up to i m if we picked columns i1 up to i m. So we've got a large number of numbers p i and these give you a point when p m plus n choose m minus 1. There's a more abstract way of doing this. Suppose we've got a vector space v contained in a vector space w where v has dimension m and w has dimension m plus n. What we can do is we can take the mth exterior power of v and this then maps to the mth exterior power of w. Well the mth exterior power has dimension equal to 1 and the mth exterior power of w has dimension m plus n m. So whenever we've got an m dimensional subspace of an m plus n dimensional space we get a 1 dimensional space inside this larger dimensional space and a 1 dimensional subspace of something just corresponds to a point of the projector space of this vector space here. In other words this is just the set of lines of this. So this is an abstract way of getting from a point of the grass manian to a point of projective space. Well of course this map isn't onto the whole of projective space. We need some plucker relations. Well the plucker relations look like this. We get that nought is sum over lambda of minus 1 for lambda of p i1 dot i m minus 1 j lambda times p j1 to j lambda minus 1 j lambda plus 1 j m plus 1. So where the columns are, where you pick m columns here and m columns here. So this is again a lot of of quadratic relations. And again this cancels out because every monomial occurs twice with opposite signs. And the proof of this is like the case when m and n are two only there's a lot more bookkeeping to do. Again, we can then check that these are all the relations you need so we need the map from g m n to the zeros of the plucker relations is onto. And I'll just very quickly sketch the proof of this. The idea, we can assume let's say p1 to m is equal to 1 by changing coordinates if necessary. Then we can find a point of the grass manian with with given values p1 octa r minus 1 r plus 1 m s. In other words at least m minus 1 indices in the set 1 to m by choosing a matrix whose left columns form the identities. But then plucker relations determine all the other p's. So that's enough to show that the map from the grass manian to the solutions of the plucker equations is actually onto. So there are several applications of grass manians. First of all, grass manians are given have a covering by a fine spaces, just as we did for g 2 2 is very similar. And these can be used to work out the co homology of the grass manians co homology groups also a product on them. The product of the grass manian co homology of the grass manians turns out to be rather complicated. It's given by something called the Littlewood Richardson rule, which I'm not going to give explicitly, but you can look it up on Wikipedia. So the Littlewood Richardson rule is really, well, it's really several different things. There are lots of different ways of looking at it, but one way of looking at it is it gives you the product of the co homology of grass manians. And grass manians also turn up in something called the line complexes. So line complexes are certain varieties with a very rich structure, for instance, the Quadric line complex is given by Well, you take the grass manian g 2 2, which you think of as being a subset of p 5, and you just intersect g 2 2 with a Quadric. So this g 2 2 is four dimensional. So if you intersect it with a random Quadric and p 5, this gives you a three dimensional variety called a Quadric line complex. And you can get things like cubic line complexes by replacing with a cubic and so on. If you've got the book Griffiths and Harrison algebraic geometry, you may notice that the final chapter is entirely about the Quadric line complex. The fourth place grass manians turn up is their quotient. So the grass manian g m n is a quotient of the group g l m plus n over K. So the group g l m plus n of K acts transitively on the m dimensional subspaces and the subgroup fixing one n dimensional subspaces a sort of block form looking like this, where these blocks are m by m and this is m by n and this is m by n and this is n by m. So they are homogenous spaces, a quotient of some group by some subgroup. Notice, by the way, that this group here is affine. It's an affine variety there for an affine algebraic group. This group here is also affine. And you might guess that if you take the quotient of an affine group by an affine group, you get an affine variety, but you don't. In general, the variety may be affine but g m n is a projective variety so it can sometimes be a projective variety. So this shows the problem of taking quotients of algebraic groups by other algebraic groups is rather more complicated than you might guess because affine modulo affine doesn't always give you affine. In fact, the quotient of two affine varieties doesn't have to be either affine or projective. So for example, suppose we take k squared minus the point zero zero. So this variety is neither affine nor projective. It's just a plain minus a point so it's not a closed subset of the plane. And there's just no way to make this into an affine variety or a projective variety. However, the group GL2 of k acts transitively on it. And the subgroup fixing a point is just a group of all matrices like this, which is just an affine line. So here we have another quotient of an affine group by an affine group and it's a bit of a mess. It's neither affine nor projective. So whatever. It's actually an example of something called a quasi affine variety and open subset of an affine variety. So another application. I forgot what number I've got up to but never mind is. It's used by growth and Dick in a construction of a Hilbert scheme. Okay, we haven't actually defined schemes yet so I'm going to have to fudge a little bit. The idea of a Hilbert scheme is it parameterizes sub schemes or projective space. Well, what does that mean well you take the coordinate ring of projective space and you look at graded ideals of this ring. So the great idea is going to be I naught plus I one plus I two will I know of course just be zero otherwise the close subset you guess is not terribly interesting. And you want to classify. Sub schemes sort of correspond roughly to graded ideals of this ring here so what you really want to do is to classify graded ideals of this ring and in particular you'd like to show that graded ideals correspond to points of some projective. Schemes. And so, and so roughly speaking you would like to be able to construct a point of projective space from an ideal like this well how are you going to do that. Well we will see later that the dimension. Of I N is a polynomial in N sorry that shouldn't be an end let's call I J is a polynomial in J for J large. It's more or less something called a Hilbert polynomial that we'll be talking about quite a lot later. So I D generates I D plus one I D plus two and so on. So, okay one problem is we have to find a D with that property which is a technical problem I'm not going to talk about then ID is a subspace of SD which is all degree D monomials. Well here we've got a subspace of a vector space. So we get a point of the grass manion of subspaces of this dimension contained in subspaces of this dimension so it's a rather large and complicated grass manion. And the grass manion is itself a projective variety and some even larger projective space so we've managed to construct points of projective space from graded ideals and this is a very large projective space obviously. I mean this grass manion is a pretty large grass manion and embedding grass manions and projects of space makes me even bigger so this is a rather huge construction. And so the next question is, what does natural mean? So what we want to do is to say the points so the lines in three dimensional projective space correspond to the points of a grass manion to two which is a certain sub variety in P five and this correspondence is natural. And what do we mean by this? I mean notice that in some sense it's trivial to find a variety whose points correspond to lines in P three. All we need to do is to take any variety with the same cardinality that we get a one to one correspondence. Well that's kind of stupid. We want to have a correspondence that makes sense. And it's not all clear how you define that this correspondence is natural. And this was answered by growth and dick. So what growth and dick did was he look at functors from commutative rings, commutative rings are. So we can take various function we can take our to the set of lines in projective space over the ring R. Or we're going to take another functor R, which takes R to G to two over the ring R. In other words, certain two dimensional projective subspaces of art before that split two dimensional or we can take our to the R valued points of say a quadric given by the plucker relation P naught one. P two three minus something plus something equals naught in P five. And growth and Dick's way of saying that the lines in P three correspond naturally to the R value points this quadric says says that these three isomorphic as functors. So this has the extra condition that if we call these functors F and G, not only a F R and F and G are isomorphic as sets. But also, these isomorphisms are compatible morphisms from R to S. So, so if we've got a key point is this suppose we've got a homomorphism rings R to S. Then we've got maps F R to F S and F R is isomorphic to G of R. And we've got a map from G R to G of S these isomorphic and the point is all these diagrams here should commute. So, in order for this to make sense, you need to work with general commuter rings are it's not really enough to work just just for projective space over fields. More generally growth and Dick showed that any scheme is defined by its functor of points. So this means that if we've got us, for instance, a scheme might be three dimensional projective space. And that means we've got a functor which takes any ring R to three dimensional projective space over ring R which is actually a little bit tricky to define we'll talk about this later. And this is a very powerful construction because you can find various functors like the Picard group of a variety or a Hilbert scheme of projective space, or an isomorphism class of a billion varieties over a ring. And these will all be funters. You can then ask, do these correspond to schemes. So the next lecture will be on a few more examples of projective varieties.