 So in this algebraic geometry lecture, we will move on from affine varieties to projective varieties and we'll start by describing projective space. So the definition of projective space over field K is you can define it as the set of one dimensional subspaces of an n plus one dimensional space over K. In other words, each line in n plus one dimensional vector space corresponds to a point of projective space. So we can use coordinates as follows. So a point of projective space can be denoted by a point of n plus one dimensional affine space. So it corresponds to the line going through this point. However, if this point is zero, it doesn't really define a line. So we have to say not all x i are equal to zero. And furthermore, if we multiply this by a scale, it defines the same line. So this point is actually considered to be the same as the point lambda x naught lambda x one and so on. lambda x n for lambda in K, lambda not zero. And so we put colons between the coordinates to indicate that we're allowed to multiply by a scale without really changing what the point is. So let's take a look at what affine what projective space looks like. Well, first of all, if x naught is none zero, we can rescale x naught and find that the point x naught up to x n, sorry, is the same as the point one y naught y one up to y n where y i is equal to x i over x naught. And this gives us a copy of affine space a to the n. So n dimensional projective space contains n dimensional affine space. And what's left over, well, if x naught is equal to naught, we're looking at the points naught x one x two and so on, up to multiplication by scalars. And this gives us a copy of n minus one, um, dimensional projective space. So in other words, we can see that projective space over a field is a disjoint union of affine space and projective space of one dimension lower. So we think of these as being points at infinity in some sense. So projective space can be thought of as taking affine space and adding some points of infinity and the points of infinity themselves form a projective space of lower dimension. So let's just look at what it looks like over the reals and the complex numbers. So let's look at projective space over real numbers. So we're looking at points x naught up to x n, not all zero, and we can rescale it. So the x naught squared all the way up to plus x n squared is equal to one just by multiplying by a suitable constant. And if we do this, there are exactly two points corresponding to the same line, which are this point here and minus x naught up to minus x n. Well, these points here correspond to a sphere s s n and these are opposite points of the sphere. So projective space of n dimensions over the reals looks like s n except we identify opposite points. So not dimensional projective space over r is just a point. It's not very interesting. One dimensional projective space over r looks like when you take a circle and you identify opposite points. And if you think about this a bit, you'll see that's just the same as a circle except it's wrapped around itself twice. So one dimensional projective space over the reals is just a circle. Two dimensional projective space over the reals is s two where we identify opposite points. And this is the simplest example of a non orientable surface. So in general, and obviously in general we do the same thing we just take a sphere and identify opposite points. What happens over the complex numbers? Well, in this case, we can take a point x naught x one up to x n of n dimensional complex space. And this time we can we can rescale so that the squares of the absolute values are equal to one. So the points with this property form a copy of the sphere of dimension two n plus one, because if we put x i equals y i plus c i. This is just the same as saying the sum of the y i squared plus the sum of the z i squared is equal to one. So we're again getting a sphere. However, we have to do more than identify opposite points, because this point is equivalent to lambda x naught up to lambda x n whenever lambda has absolute value equal to one. And the complex numbers of absolute value one just form a copy of the circle. So what we end up with is a map from s to n plus one onto n dimensional complex space, and the fibers are just copies of s one. So this is an example of something called a vibration in algebraic topology which we won't worry about what the definition of this is too much because we're not actually doing algebraic topology. So we have a vibration as being something like a product except it's kind of twisted a bit so it's not quite a product. So if we take n equals one. So one dimensional projective space over C is just a copy of one dimensional affine space plus a point, which is just the Riemann sphere, which topologically is isomorphic to an ordinary sphere s two. So what we find is we're getting a vibration s three mapping on to s two, and the fibers are just copies of s one which we denote like this so so this means we've got a base space s two s three maps on to it and all the fibers s one. So this is the famous hop vibration and algebraic topologists get very excited about things like this because if you've got a vibration. You can relate the homotopy groups of the various spaces so you can do things like show third homotopy group of s two is non trivial and so on. So another example of a vibration be s one goes to s one times s two goes to s two so this would be the projection on to s two and s three and s one cross s two are certainly not the same as each other. So, in some sense, s three. So s three is not a product of s two but it's some sort of twisted product in some sense. And of course, we can stick in higher projective spaces here and get more vibrations but we're not going to worry about that anymore. Also, you can cover projective space with copies to the cover of affine space a n. And if we look at PNC, so PNK, its coordinates are given like this. And if we take X naught not equal zero, this gives us a copy of affine space. Similarly, we can take X naught not zero and this gives us another copy of affine space and we can go all the way up to XN is not equal to zero and we get another copy of affine space. So altogether it's covered by n plus one copies of n dimensional affine space. And this is fairly typical for what happens with a projective variety. We will see that projective varieties can be obtained by gluing together copies of affine varieties. So we should think of projective space has been got by taking n plus one copies of affine space and somehow gluing them all together. So now give some historical background about where projective space come from came from. So projective geometry. Well, one of its origins was what happens if you draw a picture of something. So you've got an artist here. And he's got an easel where he's trying to draw a picture of something. And he's drawing a picture of some objects like a triangle. And he tries to draw a picture of the triangle on his picture, which means you're sort of projecting from the artist's eye from the from from this triangle to the plane that the artist is drawing on. And you can ask what properties are preserved by projection. So one property that isn't preserved is parallel lines. So for instance, the well-known example of this is suppose the artist is drawing a picture of a railway track. So here we've got a railway track looking like this. And the lines of a railway track are parallel. However, if you draw a picture of it, they're not parallel on your picture because they kind of meet at infinity. So the question is what properties are preserved? Well, straight lines are preserved because if you project a straight line, then it still remains a straight line. And people studied this and came up with a sort of collection of axioms for projective geometry. So there are two approaches to geometry. Synthetic geometry is where you write down a set of axioms. So the classical example of this is Euclid's axioms for Euclidean geometry where you have Euclid's five axioms. Actually Euclid missed out a lot of necessary axioms, but never mind. And you can also have analytic geometry. An analytic geometry, you just use coordinates and turn geometry into algebra. And we've been discussing the analytic geometry approach to projective geometry where you just write down coordinates for projective geometry and work with those. And that's what we'll be doing most of the time, but I'll just say a little bit about what synthetic geometry looks like for projective geometry. So the axioms for synthetic projective geometry look like this. First of all, any two distinct points meet a unique line. I'm missing out some background saying there should be a set of points and a set of lines and an incidence relation because I'm not actually going to be using these axioms. The second one says that any two lines in the same plane meet in one point. Any two distinct lines. There should be distinct lines and any two distinct points. Well, what do I mean by same plane? Because I haven't defined what a plane is. Well, this is a two lines meet in the same plane. If when you take four points on them and join up like this, these two lines meet. So meeting in the same plane can be defined using just points and lines. And you need a third axiom which is non-degenerate says that any line meets at least three points. And you need this third axiom because if you allow lines with two points, then there are a lot of rather stupid configurations you can have where you take two copies of something satisfying these axioms to have a line joining every point of the first object to every point of the second object. So this is a sort of non-degeneracy condition. And we can ask what objects can we get satisfying these at? You find that anything satisfying these must just be a single point which is really boring. In dimension one, we get one line plus a lot of points which is again completely boring. In dimension two, the definition of dimension two is that any two lines meet. So you find you get the axioms saying that any two lines meet in a unique point and any two distinct points lie on a unique line. So this is something called a projective plane. And there are two sorts of projective planes called desarguing ones and non-desarguing ones that we will describe in a moment. Anyway, here's an example of a projective plane. This is called the Fano plane. And it has seven straight lines and seven points. And I've drawn the seven straight lines here. And you may think one of these straight lines looks actually like a circle, but I'm decreeing that it's actually straight. So there are actually seven straight lines. So this is the Fano plane. Dimension greater than or equal to three. We can get lots of examples by just taking projective space over a field. Well, in fact, it turns out we don't need to take this over a field. We can even take it over a division ring. And it turns out these are all the examples of projective space there are. And I mentioned at least three. The only examples of projective space you get are given by doing the construction I did earlier over any division ring. The difference between one projective space over a field or a division ring is that being over a commutative field is equivalent to Papas's theorem. So in other words, you don't really gain very much generality by using synthetic geometry. Any projective space of dimension at least three, you can study using projective geometry. You can study equally well using analytic geometry and coordinates. And that turns out in practice to be an awful lot easier than trying to use axioms. So synthetic geometry using axioms has mostly died out as a research area and people only study projective space using using coordinates. Well, I mentioned that there are actually two sorts of projective planes, desagrian ones and non-desagrian ones. And these are planes that satisfy something called Des Arg's theorem. So the next lecture I'll be discussing Des Arg's theorem. Des Arg's theorem.