2
$\begingroup$

Consider a compact Riemann surface $M$ of genus $g>1$ and a divisor $D$ of degree zero. Any such divisor can be expressed as difference of two effective divisors: $D=E_1-E_2$, we can think of $E_1$ as positive part of $D$. In each linear equivalence class (i.e. divisor class), there ought to be a 'simplest' form, by which I mean a divisor $D_0=E_1^0-E_2^0\in [D]$ with deg$(E_1^0)=$deg$(E_2^0)$ being smallest in the whole class $[D]$.

For example if $D=(f)$ for some meromorphic function $f$ on $M$, then this number is zero, because $D\sim 0-0$. Furthermore, if $g>1$, any divisor of the form $D=x-y$ with $x\neq y$ cannot be further 'reduced' by linear equivalence, so the least degree of positive part for it is 1.

My question is: is there any general theory that characterizes the least degree of positive part of a divisor of degree zero?

For example, since the set of divisor class of degree zero is identified with set of line bundles of degree zero, that is the Jacobian $Jac(M)$, there should be a disjoint union $Jac(M)=Jac(M)_0\coprod Jac(M)_1\coprod \ldots$ where $Jac(M)_n$ consists of divisor class with least degree of positive part being $n$. Is any of these dense in the torus $Jac(M)$ with its usual topology? What is this partition of a complex torus look like geometrically?

$\endgroup$
1
$\begingroup$

This is the topic of the Riemann--Roch theorem, Clifford's theorem, and Brill--Noether theory.

For example, Riemann--Roch shows that if deg $D \geq g$, then there is an effective divisor linearly equivalent to $D$. This implies that if $D$ has degree zero, then we can write it as a difference of degree $g$ divisors. This gives an upper bound for your question.

In general, the morphism $\operatorname{Sym}^d C \times \operatorname{Sym}^d C \to \operatorname{Pic}^0 C$ sending $(D_1,D_2) \to D_1 - D_2$ has Zariski closed image (its source and target are projective), and so the partition you ask about is a partition of the Jacobian into Zariski locally closed subsets. Since the source of this morphism has dimension $2d$, while its target has dimension $g$, this shows directly that this morphism is not surjective if $d < g/2$. One can say much more than this, but maybe this is enough to start with.


Added:

As one more example, I'll show that if $g \geq 2$, then this map is surjective when $d = g-1$ (so any degree zero divisor can be written as a difference of effective divisors of dgree $g-1$).

We begin with a preliminary result. Namely, let $K$ denote the canonical divisor (which has degree $2g-2$), and for any effective degree $g-1$ divisor $D$, consider the linear series $|K-D|$. Riemann-Roch shows that this has positive dimension (here we use the fact that $D$ is effective), and so we may write $K \equiv D + D'$, where $D'$ is also of degree $g-1$.

Now suppose given $D$ of degree zero, and consider $D + K$. This has degree $2g-2$, which is $\geq g$ (b/c $g \geq 2$), and so we may find an effective divisor $E$ so that $D + K \equiv E$. Now write $E = E_1 +E_2$, where each of $E_1,E_2$ is effective of degree $g-1$, and then write $K \equiv E_1 + E_1'$ as in the preceding paragraph. Putting this all together, we get that $D + E_1' \equiv E_2,$ i.e. that $D\equiv E_2 - E_1'$, which shows that $D$ is linearly equivalent to the difference of effective divisors of degree $g-1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.