Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

We associate real and regular algebraic--geometric data to each multi--line soliton solution of Kadomtsev-Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non--negative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraic-geometric data for soliton data in the main cell $Gr^{TP} (k,n)$ only. Here we do not just extend that construction to all points in $Gr^{TNN}(k,n)$, but we also considerably simplify it, since both the reducible rational $M$-curve $\Gamma$ and the real regular KP divisor on $\Gamma$ are directly related to the parametrization of positroid cells in $Gr^{TNN}(k,n)$ via the Le-networks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Le--networks guarantees that the genus of the underlying smooth $M$-curve is minimal and it coincides with the dimension of the positroid cell in $Gr^{TNN}(k,n)$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $Gr^{TP}(2,4)$ and we compare it with that in Ref [3].