 This algebraic geometry lecture will cover three examples of quotients of algebraic sets. The first example is a cyclic quotient singularity. This is one of the simplest singularity forms by taking a quotient of something by a group. So we're going to take affine in a squared, whose coordinate ring is a polynomial ring in two variables. And we're going to let the group G be a cyclic group of order n, which will be generated by some sigma of order n. That's a c dot of six. And it's going to act as follows. So sigma of x is going to be zeta of x and sigma of y would be zeta of y, where zeta is just an nth root of unity. So zeta to the n equals one, we're going to take it to be primitive nth root of unity. For example, over complex numbers, we might take zeta equals e to the 2 pi i over n. Now, we want to find out what is the quotient of the affine plane by this group, except we don't mean the quotient in the sense of topological spaces. We mean the quotient in the sense of algebraic geometry. And to do that, we have to look at the coordinate ring and look at the invariance of the coordinate ring under G. So now we have to find out what are the invariance. Well, it's quite easy to work out the action of G on all the monomials because sigma of x to the y to the j is just equal to zeta to the i plus j, x to the i, y to the j. And this is equal to x to the i, y to the j, if and only if i plus j is divisible by n. So this makes it pretty obvious that the ring of invariance has a basis, consisting of the elements x to the i, y to the j, satisfying this condition here. So in order to see what's going on, it's quite a good idea just to draw pictures. Let's draw all the monomials. We've got 1x, x squared, x cubed, x to the 4, and so on. y, x, y, x squared, y, x cubed, y, y squared, y cubed, x, y squared, and so on. And now let's take n equals 3 just so that I can do everything explicitly. And we see that the invariance consists of this element here and all the things on this third diagonal. And if we go a bit further, x to the 5, x to the 6, and here we have x to the 5, y. And we'll get all the things on this diagonal and so on. So every three steps, we get all the things on that diagonal. So we can now see that the ring of invariance is generated by all the things on this diagonal. So it's generated by x cubed, x squared, y, x, y squared, and y cubed. However, obviously these are not independent. So if we call these elements c0, z1, z2, and z3, then the ring of invariance will be generated by z0, z1, z2, and z3. But obviously these are not independent, so we have to quotient out some relations between them. So what relations do you get? Well, you can see that zi, zj is equal to zk, zl, whenever i plus j equals k plus l. So we should quotient out by, well, what do we have? We have z1, z2 minus c0, z3, z1 squared minus c0, z2, z2 squared minus c1, z3. So we take this polynomial ring and we quotient out by this ideal, and this gives us the ring of invariance. So the quotient of the affine plane by the cyclic group is an affine variety whose coordinate ring is this ring here. So the next example is an example of a parameter space. So a parameter space is some sort of space whose points correspond to some configurations. A configuration isn't really a well-defined term. Roughly means it means some sort of, it might be say some sort of algebraic subset of some other variety. For example, you might have a parameter space of lines inside space or you might have a parameter space of conics in the plane and so on. We are going to look at the parameter space of cyclohexane. So if you remember from chemistry, cyclohexane is a molecule with six carbon atoms and a few hydrogen atoms that I couldn't care less about that looks something like this. It's just six carbon atoms in a ring. And what we want to do is to ask for all the ways you can arrange these in three-dimensional space. So first of all, for each carbon atom, its center is specified by three numbers. So carbon atom is just A3. So we've got six carbon atoms. So altogether, we want six copies of three-dimensional space and we get 18-dimensional affine space. So this is the parameter space for six carbon atoms. So there are some conditions these have to satisfy. So any two adjacent carbon atoms are at a fixed distance. Well, the condition for two carbon atoms to be at a fixed distance is a some sort of quadratic equation in these coordinates. So if the first carbon atom is X1, Y1, Z1, and the second is X2, Y2, Z2. So we have the condition X1 minus X2 squared plus Y1 minus Y2 squared plus Z1 minus Z2 squared is some constant. So you see this is just some sort of quadratic equation. So we have six quadrics for the distances. So that's not all we get because we also have these fixed angles for each pair of bonds. Well, the angle can be specified just by giving the distance between these two carbon atoms. So we also get another six quadrics for the angles. So altogether, we have 18-dimensional affine space and then we write down 12 equations that the points have to satisfy. So we might guess that all together that's going to give us some parameter space of dimension 18 minus 6 minus 6. We can translate it without really changing it. And we can also rotate it. So what we should do is we should take the space of dimension possibly 18 minus 6 minus 6. So although really that's only a lower bound for the dimension. Quotient it out by the group of translations and rotations, possibly reflections. And there are two translations or dependent translations and the group of rotations is also three-dimensional. So this is a six-dimensional group. So the parameter space is given by taking this space here and quotient it out by this group here. And we might, for example, try and guess its dimension. So we can ask, what is the dimension? And there's an obvious guess because here this space looks as if it's going to be six-dimensional. And we're quotient out by six-dimensional groups. So we might guess that the dimension of the quotient is going to be six minus six, which is zero. So we can ask, is it zero-dimensional? The answer is no. This was discovered by a guy called Herman Sachser who in 1890 discovered there were actually two different forms of cyclohexane. One form is called the chair form. And these look the same here. So I better move them around a bit so you can see what the difference. So the chair form, you can see these three carbon atoms here are in one plane and these three carbon atoms are another. So these three carbon atoms here are sticking out of the plane. In this boat form, these two carbon atoms are sticking out of the plane and these four carbon atoms are in the plane. So if I turn it around, you can see, if I arrange them like this, you can see the two forms are indeed different. So maybe the parameter space has two points in it. Well, that turns out to be wrong, too, because it turns out that this form of cyclohexane is flexible. You can sort of bend it round. So here I had these two carbon atoms sticking out of the plane. And if I just sort of manipulate now these two carbon atoms in the plane. So there's a one parameter space of positions it can be in. On the other hand, this form of the cyclohexane is rigid. Well, this particular one isn't rigid because these bonds are rather floppy. And if these bonds were rigid, then this would indeed be rigid. So the parameter space has, well, at least two components. It's got this point in it, and it's got a one dimensional component that corresponds to this form of cyclohexane. So this just illustrates that you have to be a bit careful about guessing dimensions of quotients. The naive guess dimension of quotient where you just subtract one for each equation you have and subtract one for each dimension of your group is a reasonable first guess for the dimension of parameter space, but it's sometimes just wrong. So the final example of today's lecture will be a modular space. So a modular space is a space whose points correspond to isomorphism classes of various things. And this sounds very much like a parameter space. And in fact, there's not really a whole lot of difference between a modular space and a parameter space. They're both essentially the same thing. However, it's traditional to use parameter space if you're classifying things embedded in something else. For instance, if you're classifying lines in three space, you call it a parameter space. On the other hand, you use modular space if you're classifying things that aren't really embedded in anything. For instance, if you're classifying isomorphism classes of elliptic curves, you'd call that a modular space, not a parameter space. If you were classifying elliptic curves embedded in four-dimensional space, you'd probably call that a parameter space. So they're more or less the same. And what we're going to do is to look very briefly at the modular space of elliptic curves. Well, one of the problems with studying this is we haven't actually defined what an elliptic curve is. So I'm just going to have to quote a few things about elliptic curves just to give you a rough idea of why modular spaces can be quotients of things. So if we look at an elliptic curve over the complex numbers, it's a nonsingular curve that's topologically isomorphic to a torus. So it looks something like this. We'll be discussing elliptic curves much more later. And we later see that any elliptic curve can be put into this form, y squared equals x cubed plus a x squared plus b x plus c, which you can write as y squared equals x minus alpha, x minus beta, x minus gamma. Well, you can obviously make some changes of variables without really changing the isomorphism class of this elliptic curve. For instance, we can translate x just by adding a constant to it and turn it into y squared equals x, x minus beta, x minus gamma. And then we can rescale x and y to turn one of these numbers into one so we get y squared equals x, x minus one, x minus lambda. It's traditional to use lambda for this parameter. So this suggests that elliptic curves might be classified by numbers lambda. However, that's not quite true because if you change lambda to one over lambda, then you haven't really changed the curve. So let's swap these two variables. So this is invariant on a changing lambda to one over lambda. It's also, if you change lambda to one minus lambda, then again the isomorphism class of this curve doesn't change. And then you can compose these so you can change it to one over one minus lambda or lambda over lambda minus one or lambda minus one over lambda. So there are six things you can do to lambda. You can change it to itself or to one of these five things. And in fact these six transformations form a group, which is in fact isomorphic to the symmetric group of order three. Let's see, did you compose any two of these transformations? You get another one similar to it. And here I should have mentioned that lambda is not equal to naught or one. So what is happening is you're getting this affine variety, which is given by a one minus the point zero or one. And you're quotient it out by this group of order six. And this space here is more or less a modular space of elliptic curves except modular spaces turn out to have larger numbers of fuzzy technical details. So that's not quite correct, but never mind. First of all, I said this is an affine variety. Well, it doesn't at first look like an affine variety because it's actually an open set of a one. Whereas I said affine varieties were supposed to be closed irreducible sets. However, if lambda is a veteran here, you can think of a one minus these two points as just the curve. Lambda times lambda minus one times mu equals naught in a two. So here we're taking lambda and mu to be covenants and a two. And you can see this curve is really just isomorph a one minus two points. So that doesn't matter. So what exactly is this space a one minus two points modulo s three. Well, we take the coordinate ring of this space here, which is just K lambda one over lambda one over lambda minus one. So it's a subring of the ring of rational functions and take it's a subring that is invariant under the group s three of order six. And in order to describe this, you want to find some polynomial in these three things that is invariant under these transformations of s three. The simplest one is traditionally called J and it's equal to two to the eight times the square minus lambda plus one cubed or divided by lambda squared lambda minus one squared. So you can check if you do any of these six transformations to lambda J remains fixed. In fact, the ring of invariance turns out to be just isomorphant or polynomials in J. So this number J is actually the famous so-called J invariant of an elliptic curve that we will be studying a bit more later on. Incidentally, this funny factor of two to the eight is made so that things work nicely in characteristic two.