Least degree of positive part in degree zero divisor class on compact Riemann surfaces

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?

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.

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 $g-1$.