 So this lecture is part of an online algebraic geometry course on schemes and will be about linear systems. So a linear system of divisors is a sort of geometric way of defining sections of sheaves used in the days before sheaves have been invented. So people were sort of implicitly working with sheaves back in the early days of algebraic geometry, but just using a different language for it. So we'll just work with, we'll take our varieties X to be non-singular projective varieties over an algebraically closed field. You can do this in more generality, but you get some minor extra complications that I don't want to think about. So we recall that a divisor is more or less the same thing as an integral linear combination. So we might take some of N, I, P, I, N, I and Z of irreducible co-dimension one sub varieties, P, I. For instance, for a curve, the P, I would just be points in a divisor is a collection of points with multiplicities where the multiplicities are possibly negative. So if you call the divisor D is effective, means all the N, I greater than or equal to zero. So it's more or less a sub variety of co-dimension one, except again you're allowed multiplicities. And now a complete linear system of a variety D naught is equal to all effective divisors that are linearly equivalent to D zero. So what this means is that it's the set of order visors D such that D minus D zero is equal to F for some function F. You remember this is the divisor of zeros, whereas usually you count a pole as being a zero of negative order. So we can rephrase this in terms of line bundles. Suppose L naught is the line bundle of D naught. So the line bundle of D naught is the one whose sections F are just rational functions whose poles are most D naught. In other words, this means F plus D naught is greater than or equal to zero. This means it's effective. So remember saying a divisor is at least zero is just the same as saying all its multiplicities are at least zero, which is the same saying it's effective. In other words, if we map any function F to D, which is F, so that should be round brackets there, F plus D naught, then this bit here is an effective divisor. So we get a map. So from global sections of L naught to divisors equivalent to D naught, so effective divisors, which is just the linear system. And furthermore, if you've got two global sections F and G, then F and G have the same divisor. It's just the same as saying F is equal to lambda G for lambda some nonzero element of K. So what we do is we get the set of the linear system of D naught can be identified with the projective space of all global sections of L naught. So this is a vector space of global sections. And this just means you take the corresponding projective space. So this shows that linear systems or complete linear systems are more or less the same as taking all sections of a line bundle, except that it is dimension one less because the linear system is really just a projective space of the corresponding line bundle. So let's look at some examples of this. First of all, let's take X to be P1. And let's take the divisor D naught to be n times the point of infinity. Then we want the linear, then the corresponding line bundle L naught is isomorphic to O of n. And the sections just correspond to polynomials a n X to the n plus plus a naught of degree less than or equal to n. And the zeros of this polynomial, which corresponds to linear system. So the linear system is just all effective divisors of degree n, which just means collections of n points. The points are allowed to be the same. You kind of points with are allowed to have multiplicities. And so that's a particularly simple example. You can do the same thing for X being P to the n. Again, we can take the divisor to be n m times the points at infinity. Again, this will correspond to the line bundle O will be isomorphic to O of n. And the sections correspond to homogeneous polynomials of degree n. And the linear system corresponds to all sort of corresponds roughly to all hypersurfaces of degree n. So a degree m, I guess, except that's not quite correct because you allow sort of degenerate hypersurfaces. They're allowed to be reducible and they're allowed to have components with multiplicities bigger than one and so on. So saying they're all hypersurfaces should be taken with a with a small grain of salt. Next, we can have the complete linear system of divisors corresponds to all sections of the corresponding line bundle. More generally, we can have incomplete linear system, which might correspond to some subspace of sections. So here they the element of a linear system corresponds to the zeros of a section. So here we're just looking at the zeros of some subspace of sections. For example, let's take X to be the two dimensional projective plane. And let's take D naught to be the line at infinity. Now here the global sections of the corresponding sheaf will just be rational functions with at most a pole of order one at infinity. They're allowed to have a pole on D naught. So it's spanned by one X and Y. Here we're taking coordinates X, Y, Z for the projective plane. And if you can find them to the obvious affine plane in P squared, that the global sections of this line bundle are just all linear functions. Now, if you take the complete linear system correspond to this, you just get all lines in P2. So let's take the subspace just spanned by X and Y. So this is going to be an incomplete linear system. So this will be all linear functions of form AX plus BY and the zeros of AX plus BY just look like this. So they're just going to be all lines through the origin. So this is an incomplete linear system. So a typical line might be Y minus 2X. So this would be some section of the space. And this would correspond to this line here of the linear system Y equals 2X. And what you notice is that this linear system has a base point. So a base point is a point in all divisors of the linear system. Well, a base point just corresponds to a common zero of all sections in the subspace you've chosen. Now, you remember that if we had a collection of sections of a line bundle, it gave rise to a map to a projective space. And this map was well-defined on the whole of the variety if and only if the sections generate the line bundle. So you can see that the sections generate the line bundle just means that for each point, there's some non-zero section of the line bundle. So sections generating the line bundle is equivalent to saying there are no base points. So if we're working with line bundles, we say we get that the map to projective space is well-defined everywhere, provide the sections, generate the line bundle. If you translate this into a language of linear systems, it just means the linear system has no base points. So there's no base point that's a zero of all the divisors or equivalent that's a zero of all the sections you've got there. So in this particular case, we get a map from P2 to P1, except it's not actually defined everywhere. So here we get a map from P2 to P1, and it just takes the point x, y, z to the point x, y, and you see it's not defined at the base point, which is 001. And so in general, if you've got a linear system of divisors, you get a morphism from some open set of your space x to projective space, where the projective space is actually the dual of the projective space of the linear system. So far, we've just been looking at linear systems on projective space, which are particularly easy because projective space has vanishing Picard groups. So let's look at an example with, sorry, not non-vanishing Picard group, vanishing Picard variety. So let's look at an example with a non-vanishing Picard variety. And here I'm going to take x to be an elliptic curve. And as usual, we're going to cheat a bit by doing it over the complex numbers so that we can use analysis and elliptic functions to see what's going on. And we're going to look at some linear systems of divisors over the elliptic curve. So there are two ways to think of the elliptic curve. You can either think of it as being C modulo lattice, which is a sort of analytic way of doing it, or you can write it as a curve y squared equals 4x cubed minus g2, g2x minus g3. And the connection is that x and y correspond to elliptic functions. So you remember there's an elliptic function p of z, and we have its derivative p dash of z. And y is equal to the derivative of the elliptic function, and x is equal to the elliptic function p of z. So this just becomes the usual differential equation for the viastric elliptic function. I can just write it out explicitly just to make it clear what's going on. So that is equal to p of z cubed minus g2 p of z minus g3. So now let's look at a few divisors. First of all, let's take d0 to be a point, one times the point zero. So if I draw the lattice for the elliptic curve, so here's some lattice contained in the complex numbers. Here's the point zero. And I'm just taking my divisors to consist of a single point here at zero. Then we can ask what are the sections of the corresponding line bundle? Well, the sections of l0 just consist correspond to elliptic functions with at most a single pole at nought. And the only elliptic function of this property is constants. So the sections of the corresponding line bundle just correspond to the constant elliptic function. So the linear system of this is just d0. So the linear system is nought dimensional and just consist of a single point. So for the projective line, if we had a single point, we can sort of move it round on the line and get a linear system of dimension one. For elliptic curve, we can't move a point round. It's not linearly equivalent to the point somewhere else. So the linear system just consists of a single point. And we notice that it has a base point consisting of this point nought because obviously if the linear system just consists of the point nought, then nought is a point of all the divisors of the linear system rather trivially. So now what happens if we take a divisor to consist of two copies of this point zero? We're going to take d nought to be two copies of nought. Well, now we get more functions in the corresponding line bundle because we can get the function one and we can also get the vice tricelliptic function which has at most a double pole here. So we get all functions of the form a plus b times p of z. And now we can ask what are the corresponding divisors corresponding to this? Well, the divisors consist of a point plus minus the point. So if you've got an elliptic curve here, a typical divisor on this linear system will consist of a point here and minus the point or we might get another pair of points of the linear system. We might have a point over here and a point over here and so on. So what do these look like in the algebraic curve? y squared equals 4x cubed minus g2x minus g3. So the elliptic curve might look something like this. Well, what they consist of is pairs of points with the same x coordinate and they have the same value of p and the same value of one. The pair of blue points might correspond to a pair of blue points there and the pair of red points might correspond to a pair of red points somewhere else and so on. So they just have the same coordinate. You notice by the way there are three double points in this linear system. So there's one double point here. Let me use a color that's different from red. There's a double point here and there's a double point here and a double point here and a double point here. So at each half lattice point there's actually a double point and these correspond to these three points on the elliptic curve because that's where the vertical line intersects the elliptic curve in a double point. So now let's see what happens if we take d to be three times the point zero. So this time the linear system is going to consist of point z1, z2, z3 such that z1 plus z2 plus z3 equals zero. So we're working inside c modulo of a lattice L. So what does this look like for the elliptic curve? Well, let's think about what we're doing. The corresponding space of sections of line bundlers, three dimensions spanned by one p and the derivative of p. So we're working at points with a plus b times p plus c times p prime equals zero. So z1, z2 and z3 are going to be the zeros of this function. So this vanishes at z1, z2 and z3. Well, this just says a plus bx plus cy equals zero. So the linear system just consists of triples of points lying on a straight line. So here's a straight line and this would be, this might be z1, z2 and z3. And any other line, you can draw any other straight line like this one here. And again, these three points will be an element of the linear system. So in this case, the linear system consists of triples of co-linear points on the elliptic curve. You can also ask what happens if instead of asking for z1 plus z2 equals z3 equals 0, we ask for z1 plus z2 plus z3 equals something else. So now let's take d0 to be, say, 0 plus some point z. Now the linear system will consist of point z1, z2, z1 plus z2 plus equals z, or z1 plus z2 plus minus z equals 0. Again, this is in c modulo l. So what does this look like on the elliptic curve? Well, if we draw the elliptic curve and if we pick a point, if we pick some fixed point here, then the linear system is just going to consist of pairs of points on the line through this. So this pair of points will be on the linear system, or this pair of points will also be on the linear system. So the linear system is going to be pairs of points, say RS such that RS and T are collinear for some fixed point T. So what this shows is that the elliptic curve has continuous families of linear systems. So for projective line, we only got a discrete set of linear systems. So the linear system was determined by the degree of the points in it. But for elliptic curves, we've got a continuous family of different linear systems because the point z varies continuously. Okay, that's all I want to say about linear systems. So next lecture, we'll be looking at a very general construction which generalizes both the construction of projective space and blowing up points.