For this whole post, we’ll take to be a curve and the Jacobian of the curve. We’re going to construct several special subvarieties (not special in any technical sense, though) of , which encode a great deal of geometric information about .
We start with a problem: is there a natural map ? The answer is no, sadly. However, all hope is not lost! Nay, there IS a natural map , given by taking each point to the divisor represented by . For , this map is an isomorphism onto its image (injectivity is easy, as two points being linearly equivalent implies genus zero, smoothness of the image is a bit less obvious, but not hard). So we know that sits inside of , but that requires some identification of with .
In fact, for all , we have a map given by taking . This even factors through , because the addition on is abelian. We will call these the level Abel-Jacobi maps, and if , we call it the Abel-Jacobi map, and denote them by , and drop the subscript for . We call a curve in an Abel-Jacobi curve if it is of the form where .
Moving to higher dimensions, we denote the image of by , which is the locus of effective divisors in . For any , the fiber is the set of all divisors linearly equivalent to it, and thus is a projective space, naturally identified with .
We can define a function , which is just the dimension of the fiber over . This function is upper semicontinuous, that is, the loci where are all closed. We define these loci to be , that is, is the locus of all complete linear systems of degree and rank on . We’ll return to these later. For now, we’ll study .
Here, the codimension is 1, so we have a divisor, as is a smooth variety. The Riemann-Roch formula tells us that for , we have , so is an involution of , which we’ll denote by . We’ll use the same letter for the involution applied to arbitrary points of . Now, if we take a symmetric theta divisor, that is, as before, such that if then , for , then we get a nice theorem:
Riemann’s Theorem: There exists a constant such that .
This theorem gives us a geometric model of the theta divisor, which we can now construct explicitly from the curve. It’s the first step in a program that ties together the geometry of the pair with the geometry of , and culminates in the Riemann Singularity Theorem and the Torelli Theorem. We’ll pick up these threads later.
For now, we’re going to look more closely at . If we denote translation by as , then we can write Riemann’s Theorem as . Now, looking at , we note that symmetry of , Riemann’s Theorem, and the fact that is -invariant, tells us that , and so . Thus, is a line bundle satisfying . Such line bundles are called theta characteristics, and these turn out to be extremely closely tied to the geometry of the curve .