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?