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 GrTNN(k, n). In ref. 3 (Abenda, Grinevich, CMP 2018) we were able to construct real algebraic–geometric data for soliton data in the main cell GrTP(k, n) only. Here we do not just extend that construction to all points in GrTNN(k, n), but we also considerably simplify it, since both the reducible rational M-curve and the real regular KP divisor on are directly related to the parametrization of positroid cells in GrTNN(k, n) via the Le-networks introduced in ref. 63 (A. Postnikov). 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 GrTNN(k, n) to which the soliton data belong to. Finally, we apply our construction to soliton data in GrTP(2, 4) and we compare it with that in Ref. 3.