Ok, so last time, we discussed divisors. We’re going to keep going in that direction now, and now we’re going to talk about linear systems of divisors. Whenever we talk about linear systems, we’ll assume that our variety is nonsingular, so we can even talk about Weil divisors with no problem, though we’ll sometimes also use Cartier divisors due to how things will be handed to us.
Last time we took a Cartier divisor and got a line bundle. So let a divisor and the line bundle associated to the divisor. Now take . On open sets where is trivial, restricts to a regular function, so we get a Cartier divisor which we will call . This can also be taken as a formal sum of codimension one subvarieties. Now, the following are true, but I won’t prove them:
- For any which is nonzero, the divisor is linearly equivalent to and is effective (recall that that means that the coefficients of the Weil divisor are all nonnegative)
- Every effective divisor linearly equivalent to is for some .
- If have , then there is a nonzero such that .
Now, this tells us that the vector space of global sections of , minus the origin, maps down to the effective divisors linearly equivalent to , but that nonzero scalar multiples get identified. This should sound familiar…mostly because it’s precisely how projective space itself is constructed. That means that there’s a natural structure of projective space on the set of effective divisors linearly equivalent to a given divisor!
So we define a complete linear system to be the set of all effective divisors linearly equivalent to some given divisor , which forms a projective space, and is denoted . A linear system is then , a linear subspace of , so it just corresponds to a vector subspace of . We define the dimension of a linear system to be the dimension of the projective space it defines.
Now, a bit more terminology: a point is a base point of a linear system if and only if for every , we have in one of the prime divisors of . In terms of sheaves, this says that for all , where is the vector space determining , we have .
Remember when we talked about how a line bundle determines a rational map to projective space? Well, in truth, it might define quite a few, though there is one into a biggest projective space arrived at this way. In fact, a morphism is the same as a linear system without base points on and a set of elements in the vector space determining it which span it.
So now when is the map a closed immersion, as defined in the comments on the post on line bundles? The conditions are that the linear system separates points and tangent vectors. The first condition is that for all points , we have a divisor in the linear section so that is contained in one of its prime divisors but is not. That is, there’s a function which is zero at one point and nonzero at the other.
Separating tangent vectors is a bit more mysterious. Take and latex . Remember that this is just a linear map . Then we want there to be a divisor such that . This tangent space makes sense, because is effective, and so gives an algebraic subset of , so . The point is that for any tangent vector, we can have it point in a direction not along some divisor.
Now that we have this definition, let’s do some examples. We’ll take , and choose our divisor to be where is a hyperplane and . It’s a fact that every divisor on projective space is linearly equivalent to one of this form. So now we look at . This will consist of all formal linear sums of hypersurfaces with . These are precisely given by the homogenous polynomials of degree . We can choose as a basis for the complete linear system the functions given by monomials of degree . Now, this system is base point free and separates both points and tangent vectors, so we get a morphism , the latter dimension being the dimension of the space of homogeneous degree polynomials after projectivizing. We’ve actually already seen this map! It’s just the Veronese Embedding, phrased with linear systems. A similar construction can be used to get the Segre Embedding, or, as mentioned, any map into projective space.
That seems enough for now, and we’ve got a couple of options of where to go next that I’ve been thinking about, and I’m going to leave it up to you readers. Post a comment to let me know which of the following is preferred, and I’ll do it:
- Riemann-Roch Theorem and the geometry of curves
- Bertini’s Theorem and more about divisors, including generalizing to cycles and some intersection theory
- Something rather different: some computational techniques, blow-ups, and the 27 lines on a cubic surface
- Other suggestions? I know what I’d do for the other three, but if something is suggested that people want to see me make an attempt at explain, I’m open to the possibility