 this algebraic geometry video will be about toric varieties which give an easy way to construct several examples of projective varieties. So we'll start with an example of how to construct affine varieties from cones. I will do the two-dimensional case as an example. Let's take the ring generated by x x minus one y y minus one. So this is a coordinate ring of k star times k star where this is the variety of non-zero points in k. Well, strictly speaking, the non-zero points in k don't form an algebraic set, but we can think of them as being the points on our hyperbola or something like that. So it's convenient to think of them as being the non-zero elements of k. And I'm going to draw a picture of this ring by drawing a point for each monomial. So what happens is I get a sort of lattice like this and the various monomials are one x x squared and then we get y y squared x y. And of course, down here we get y to the minus one x minus one and so on. So each point of the square lattice corresponds to some monomial of this ring. And now I can define subrings just by drawing cones like this. So here is an orange cone and I'm going to take the subring generated by all monomials in this orange cone and it's pretty obvious what that is. This is just going to be the polynomial k x y. So that wasn't a very exciting example. Let's try another one. Suppose I take this green cone, then this is going to be all polynomials in x and x y to the minus one. Well, this is just isomorphic to k x z where z is equal to x y to the minus one. So again, this cone just gives us a polynomial ring in two variables. Now let's try this cone here. Well, it looks at the first site as if this is going to give us yet another example but something different happens because you see if we take these points they correspond to x y to the minus one and x minus one y to the minus one. If we take these two monomials here they do not generate all points in the cone. They don't generate the monomial y's. So here we're now getting the ring k x y to the minus one y to the minus one x minus one y to the minus one. And let's try and identify this as an abstract ring. Well, let's write this as k. Let's call this element a, this element b and this element c. Well, there must be some relations between a, b and c. In fact, you can easily see that b squared is equal to ac. So we should quotient out by the ideal generated by b squared minus ac. And the set of points in three space where b squared equals ac, it's quite easy to sketch. It just forms a sort of double cone with a singularity at the origin. So whenever you take a subring given by a cone the subring has the following properties. First of all, it's an algebra over k. Secondly, it has no nil postant elements. Which is obvious because it's just a subring of this ring here. Thirdly, it's sometimes finitely generated. Well, it turns out it's finitely generated if the cone is rational, meaning in the two dimensional case that its edges pass through a rational point. If you take a cone with an irrational edge it's not too difficult to see it. In fact, it's not finitely generated. It still gives you an algebra but it's not a finitely generated algebra. So it doesn't correspond to an affine variety. So if it's finitely generated then what we get is the coordinate ring of an affine variety. And it's easy to see what the affine variety is in these simple cases. In this case, this orange cone corresponds to the variety a squared. In this green case, it again corresponds to a squared even though it looks smaller. And in this case, it corresponds to a cone, a double cone, whatever. Even though this right angle quadrant looks superficially like this right angle quadrant. So we get a map from rational coat but we get an affine variety associated to every rational cone. Now let's look at what happens to varieties associated with various cones. So suppose I take two cones. Let me take orange cone and a bigger red cone. And we can ask what is the relation between the corresponding varieties? Well, obviously we get a map from the orange ring to the red ring. In fact, it's a subring of the red ring. Well, we now run into a rather annoying problem which is that if you've got a map from a coordinate ring to another coordinate ring, the map between corresponding varieties goes in the opposite direction. So we get a map from the red variety to the orange variety. And this is kind of annoying because we've got a map from the orange cone to the red cone. But instead we get a map from the red variety to the orange variety, which is the wrong way round and very annoying and confusing. So we can get round this as by using duality. So if we take a lattice z squared, we can take it's dual lattice. Well, the dual of z squared of course looked rather like z squared. And now, if we take any cone in z squared such as this cone here, we can now look at the dual cone, which consists of all points in the dual of z squared that are positive on this cone here. So the dual cone of this will now look something like this. And now, if a cone is contained in another cone, the dual cone, the first one will actually be bigger than the dual cone of the second. So by taking duals, we sort of reverse direction of morphisms. So the idea is instead of associating a variety to a cone, we associate a variety by taking the ring associated with its dual cone. So what we're doing is for each cone in z squared, look at the dual cone and take the corresponding ring. So, and if we call this cone C, so this gives a variety of the cone C. So let's just see what happens in a couple of cases. So suppose we take a cone to be a sort of degenerate cone. So I'm just going to take a cone to consist of a single line. And we can ask what is the dual cone? Well, the dual cone is now the dual of this line, which is in fact an entire half plane. And the coordinate ring of this will be the ring K X X, the minus one Y, which corresponds to the non-zero elements of K times K. So a single line corresponds to the variety K star times K. On the other hand, if we take a quadrant like this, then the dual of this quadrant will again be the same quadrant. And this just corresponds to the ring K X Y, which corresponds to an affine plane. So the quadrant corresponds to the plane K times K. Now you can see that K star times K is a sub variety of K times K. And at the same time, the cone is a sub cone of this blue cone here. So by taking associating varieties to dual cones, we get the same relation between cones as between varieties, which makes it a lot easier to figure out what's going on. Well, now, instead of looking at just one or two cones, we can look at several cones. So what we can do now is take several cones like this. Let me first do a one-dimensional case. So I'm just going to take one-dimensional case and I'm going to take some cones. So I'm going to take a red cone, which is just going to be a line, and a blue cone, which I'll take with the opposite line, and I will take the intersection, which is a green cone and just corresponds to a point. And now I should look at the dual cones and see what's going on. So the dual of the red cone is pretty obvious. It's just this line here and the dual of the blue cone is again just this. And the dual of the green cone now becomes the whole line. And we can work out what the varieties are corresponding to this. So the red cone K of X just corresponds to an affine line. The blue cone again is another affine line. But now the green cone corresponds to KXX minus one. Which is a one minus point. And now what I'm going to do is I'm going to take these three varieties. So I've got two copies of a one and a one, and they both contain a copy of a one minus naught. And now I'm going to glue them together. So the picture is I've got a red variety here, which is a copy of a one. And I've got a blue variety here, which is another copy of a one. So this is an a one and this is another a one. And they're glued together along this green sub variety, which is a copy of a one minus the origin. Well, what do we get? Well, it's not very difficult to figure out what you get. What you get by doing this is just a copy of the projective line P one. So in other words, we can picture P one as being the union of two copies of a one glued along a copy of a one minus a point by drawing this picture here. So, so this green one is the a one minus point. And this red one is corresponds to the a one and this blue one corresponds to a one. Well, of course, in one dimension, that's not terribly exciting. In two dimensions, you have a lot more room to do things. So let's see what we get. Well, first of all, let's try the most obvious thing we can do and just take four quadrants. I'm going to take a green quadrant, a red quadrant, blue quadrant and some other color quadrant. Let me take a purple quadrant. So what we end up with is four varieties, each isomorphic to a two. There's a two there, there's a two there. And we're gluing them along these various sub varieties. So what does this correspond to? Well, this one corresponds to an a two minus a copy of a one. This is essentially just a one times a one minus the origin. Well, what are we getting? Well, it's not too difficult to see. This picture is really a product of two copies of the picture we had on the previous piece of paper. And so if you take all these copies of a two and glue them together, what we get is not p two as you might guess, but p one times p one. So this is a picture of the projective line times the projective line as it's called the varieties we get like this sometimes called Toric varieties. So p one times p one as a Toric variety is drawn like this. Well, how do you get p two? Well, we can get the projected plane p two like this. So I'm going to take a copy of p one here. And then I'm going to take another, sorry, this isn't a copy of p one, it's a copy of a two. And then I'm going to take this red cone here. And what does this correspond to? Well, we take a quick look at what it's dual cone is. So the dual cone of this red cone will essentially be this thing here, which as we saw earlier is just a polynomial ring in two variables and corresponds to a two. So, so this cone here gives us a copy of a two. And finally, we can take a blue cone here, which of course gives us yet another copy of a two. So we now have three copies of a two and they glued on these two. Three p two because p two is a copy of three copies of the affine line glued along copies of a one. Well, we can do other things. For example, we can take the following cone let's take the cone here, cone here, cone here, cone here, and the cone here. So this has got by gluing together five copies of a two along something or other. So what's this? Well, you can go and figure it out for yourself. Well, we can do even more exotic things. For example, suppose I take a cone that looks like this and then I take another cone that looks something like this and then I take another cone that looks something like this and I take another cone looking something like this. As you can see, what I can do is I can take an infinite sequence of cones and I can glue together a lot of copies of two dimensional affine space according to this infinite sequence of cones. And the question is what projective variety do I get from that? And the answer is I don't get a project variety. Variety is too big to be projected. In fact, it's not quasi compact and all projective varieties or open subsets of them are always quasi compact. So what this gives is an abstract variety. Listen, if I have a finite number of rational cones then I get a projective variety, but it's not all that difficult to check this. So the sort of varieties you get like this are called Toric varieties. The reason is that they all contain a torus as a dense sub-variety. So first of all, I'll explain why that is so. And then I explain why a torus is called a torus. So if I take any collection of cones like this, we notice I can just look at this single point here and the single point will sort of be contained in all these. So the variety of this point will map to whatever abstract variety I've got here. Now the variety of a point, to get the variety of a point we take it as dual cone, which is just the whole space. And the whole space corresponds to the ring KXX the minus one Y, Y to the minus one, at least in two dimensions. And this thing is the coordinate ring of a torus. So I'd now better explain why this is called a torus. So in algebraic topology, we know what a torus is. It's something that looks like this. So it's S1 times S1. And more generally, the product of any number of copies of S1 is also called a torus. Well, let's try doing this in algebraic geometry. So we can form S1 over the reals. So S1 over the reals is going to be the set of points with X squared plus Y squared equals one. So what happens over the complex numbers? Well, over the complex numbers, this looks like X plus IY times X minus IY equals one. And by changing variable, we can just say Z1 times Z2 equals one. This is just a hyperbola, which over the complex numbers is more of the complex numbers with the origin removed. Just map. Z1 can be any non-zero complex number. So in other words, we can think of the non-zero complex numbers as being a sort of analog of the circle S1 over the reals. So the circle over the reals corresponds to the non-zero complex numbers over the complex numbers. So more generally over the complex numbers, we can call the product of any number of copies of C star a torus in the same way that the product of any number of copies of S1 over the reals is called a torus. In the theory of algebraic groups, a torus of this form, a product of copies of the non-zero complex numbers plays the same role that an ordinary torus plays in the theory of compactly groups, which is another reason for calling these both toruses. So that's what a toric variety is. It's something you can write down amazingly complicated toric varieties just by drawing pictures of several cones in dimensional space. And they all have a torus like this as some sort of dense subset.