Divisors on rational M-curves

If one applies degenerate spectral curves for constructing solutions of soliton equations, it is necessary to refine the definition of the divisor, because if a pair of divisor points approaches a double point from different directions, the Abel transform remains finite, and it is necessary to apply resolution of singularities procedure to make the dynamics prescribed by the equation continuous. It is very likely that in generic situation highly complicated singularities may occur. We show that in the special important case of real regular mutlisoliton solutions of the Kadomtsev–Petviashvili II equation the situation is essentially simplified. If we consider the rational M-curves, corresponding to the totally positive Schubert cells, and divisors, satisfying the Dubrovin–Natanzon conditions, only two simplest types of blow-ups take place.