 So this algebraic geometry lecture will be mostly about de-singularizing an algebraic curve. So this arises because, as mentioned in an earlier lecture, function fields that are finitely generated over a field and of transcendence degree one sort of correspond to non-singular projective curves. Hot Sean gives a way of getting from a function field to a non-singular projective curve by constructing the curve directly from the function field. We're going to use a different way of doing this. First of all, if a function field is transcendence degree one and is finitely generated, then it's generated by two elements X and Y with X being transcendental over K. And there's some polynomial relating X and Y because Y is algebraic over this. And this defines an algebraic curve and you can easily make it into a projective curve. So this gives you a projective curve, possibly singular. So the problem is how do we get rid of the singularities because then we will get a non-singular projective curve. So what we want to do is to show how to resolve singularities of a curve. And the method we're going to use is sort of due to Isaac Newton except he didn't stated in terms of desingularizing curves. He stated in terms of finding a sort of expansion of solutions of algebraic equations using a kind of power series. And we're going to show we can resolve singularities of the curve FXY equals naught by repeated blow-ups. So what we'll do is we'll have a very simple algorithm. Whenever there's a singularity of the curve, we just blow up at that point and just keep on doing that and we'll show that that eventually produces a non-singular curve. We're going to assume the characteristic is equal to zero. This proof doesn't actually work in characteristic P although the result is still true in characteristic P greater than zero. So the first thing you do is we're going to change coordinates. So the singular point, we're going to assume the singular point is zero, zero in a fine space. And we look at the lowest degree terms. The lowest degree terms are going to look like something like ANY to the N plus AN minus one Y to the N minus one X plus A naught X to the N plus terms of degree greater than N that we don't really care about. So N is called the multiplicity of the singularity. And this, the lowest degree terms form a homogeneous polynomial. So it has roots are in one dimensional projects of space. And what these roots correspond to is kind of the directions of the singularity. What I mean by this is suppose we've got a singularity. It might have some bit coming in like that and it might have another bit coming in like that. So you can think of this singularity as a sort of having two singular curves going off in this direction here. So and one singular curve going off in that direction there. Now the lines through the singularity give you a copy of one dimensional project of space. So the roots of this homogeneous piece just sort of tell you what direction the various bits of the singularity go off in and what their multiplicities are. And blowing up the point means you kind of separate out the different directions. So what would happen for a singularity like this with kind of two lines coming in with that direction and one line coming in with that direction. When you blow it up, you would separate out the line in this direction from the lines in that direction. So this singularity might have multiplicity three and when you blow it up, you get something of multiplicity one corresponding to this line which is non-singular and something else of multiplicity two corresponding to this line here. So what you see is one blow up reduces the multiplicities unless all n points of p one are the same. In other words, the degree n terms a n y to the n plus a naught x to the naught must be equal to a n y plus alpha x to the n assuming a n is non-zero. Let's take a n to be non-zero. So in this case, there will be the singularity of n lines all going on in the same direction. If you blow it up, we will still have a singularity of multiplicity n. So well, if it reduces the multiplicity then that's fine. We've made the singularity simpler and we can keep going. So we can now concentrate on the case where all n roots of this homogeneous polynomial are the same. Now we can make a change of variable. We just change y to y minus alpha x and the degree n terms become just y to the n. So we'd have y to the n plus terms of degree greater than n. So what happens now? Well, let's look at an example. Suppose you've got something like y squared plus x to the nine plus y to the five x cubed. Let's think what happens if we blow it up. Well, blowing up this point at nought corresponds to changing either x to x times y, which you can easily check. Doesn't produce any singularities. Or we change y to y times x. Remember, blowing up, we have to cover p one by two copies of affine space and do some coordinate change and the coordinate change turns out to look like this. So then it becomes y squared x squared plus x to the nine plus y to the five x to the five x cubed equals zero. And we can now take out the factor of x squared and get y squared plus x to the seven plus y to the five x to the six equals zero. And you look at this and it's not entirely obvious in what way this has improved. You see the term y squared has stayed the same. That's fine. The term x to the nine has become x to the seven, which is arguably better and the term y to the five x cubed has become y to the five x to the six, which is arguably worse. So it's not at all clear in what sense blowing up has improved this singularity. Well, let's draw a picture of what's going on. What we're going to do is to draw something that is more or less the Newton polygon. What we're going to do is we're going to draw a square quadrant of points and we're going to have each point correspond to some monomial. So this will be one, this will be x, this will be x squared, this will be y and so on. And up here, we have a term y to the n corresponding to the smallest degree monomial in our singularity. So here are the monomials of degree n and the only one is going to be y to the n. And then there are going to be some other monomials of other degrees, for instance, we might have a y to the nx squared up here and so on. So we have the various other monomials. And the question is what does blowing up do to these monomials? Well, if we have y to the n times some power of x, you can see that's not going to do anything because we're going to change y to the n to y to the nx and then divide by x to the n again. So all these monomials kind of stay put. Well, what about something like y to the n minus one x cubed? Well, we change y to y times x. So we get y to the n minus one x to the n minus one plus three and then we take out the factor of x to the n, which gives y to the n minus one x to the three minus one. So what happens is monomials here, just kind of go one step to the left. And similarly, monomials down here go two steps to the left. So if you've got a factor of say, x to the seven times y to the n minus two, it becomes x to the five times y to the n minus two on blowing up. On the other hand, monomials up here get a bit worse. So if it's in terms of the form y to the n plus one, something the power of x will increase. So blowing up kind of shifts all monomials with whose power of y is less than n to the left and shifts all monomials whose power of y to the n is bigger than n to the right. You can see this happening. So x to the nine is down here and gets shifted to x to the seven and so on. And now we seem to have finished because we can just keep blowing up and we shift all the monomials to the left until at least one of them hits this line. And then we're back in the previous case where we can reduce multiplicity by blowing up. Well, that almost works, but there's one problem. So we say blow up until some monomial is on the line y to the n minus i x to the i. So as soon as we've got a monomial on this line, then we can blow up and reduce the multiplicity. Well, there's a slight problem. If we do this, the monomials of degree n might be another nth power. And then you see the problem is we're going around in circles because we start with some monomials on this line of monomials whose power is n. And then we blow up. And as long as we don't have an nth power, we reduce the multiplicity. If it isn't nth power, then we shift monomials to the left until some of them are on the line. But then if it's an nth power again, we just seem to be going round and round in circles. We keep on making change of variables to make this nth power a power of y and then blowing up to move things to the left. And it's not at all clear that this process ever terminates. Well, let's think about what's going on. So if we've got our monomials like this, so here's y to the n. Now, the first time we blow up, we might shift everything to the left. We might shift this one to the left and then blow it up. Well, we can get, getting rid of this term corresponds to making the linear transformation y goes to y plus something times x squared. So that will sort of get rid of the term here, where this is y to the n minus 1 x squared. The next time we go through this cycle, the problem is caused, we might have a problem caused by something here. So we would then try and make the transformation y goes to y plus something times x cubed and so on. So let's try and think about what happens. What we do is we keep on applying transformations y goes to y plus something times x to the n for n being equal to two, three, four, five and so on. And there are two possibilities. First of all, this stops. Then what that means is eventually we get some sort of expression where this term, we've got a term in y to the n and this term here is zero. So the term in y to the n minus 1 x is zero and there are some non-zero terms here. So we can all, so we can keep on making this term here by making these substitutions y goes to y plus x to the n. So we can, every time we cycle through this process, we can always arrange that this coefficient here is zero and possibly, but there may possibly be non-zero terms down here. Well, if this term here is zero and the terms down here are non-zero, then we can stop because y plus alpha x to the n is equal to y to the n plus n alpha x to the n. So the n is equal to y to the n plus n alpha y to the n minus 1 x and so on. And this term here is non-zero if the characteristic is equal to zero. So this is where we use the fact that the characteristic of the field we're working over is zero. So this shows that either we go through this process an infinite number of times or it must eventually stop and we get something that isn't the power of a linear term in which case, blowing it up one more time will reduce the multiplicity. Well, there's one more problem. What if we get an infinite number of substitutions y goes to y plus something times x to the n. In other words, whenever we keep doing this, we never end up with, if we make, we keep on making this coefficient zero and what happens if these coefficients are never non-zero. Well, if this happens, what we do is we find an infinite power series. The substitution y goes to y plus something times x squared plus something times x cubed and so on transforms our original polynomial to a power series of the form y to the n plus powers plus terms in y to the something that's greater than or equal to n. And what this means is the original polynomial is divisible by some power series to the power of n where this number n is greater than or equal to two. And this really can happen sometimes. For example, if our original polynomial was y squared minus y plus x squared or squared. So this is the equation of a kind of circle passing through the origin except we've squared it. Then you find if you take this polynomial f x y and keep blowing up points, then you never do resolve the singularities. Well, the reason for this is that this particular polynomial has singularities at every point because it's a power of some non-trivial polynomial. You see, if our original polynomial was divisible by some positive power of a power series, then if we test for singularities, we find it would actually be singular at every point. So this is not possible as the original polynomial f x y has no multiple factors. In fact, we're assuming it's an irreducible curve. So it has no factors at all, but factors don't really matter. What causes problems is multiple factors. In fact, as I mentioned, this process really does break down if you allow multiple factors because if you're defining an algebraic curve, it doesn't have multiple factors, but you can, as we'll see later, we can define sub-schemes defined by a polynomial with multiple factors. And for these sorts of sub-schemes, you cannot resolve singularities by repeatedly blowing up points. They have singularities at every point and you can blow up points until you're blue in the face and they will remain non-singular. So this process gives an algorithm for resolving the singularities of polynomials, provided the polynomial has no multiple factors. So I mentioned this proof was sort of due to Newton. Newton actually proved something slightly different. He showed that you could find something called a Prisa expansion for any algebraic function in terms of something called a Prisa series, which are named after Prisa, but are in fact originally invented by Newton. So I'll say something about that in the next lecture.