Parametrization of positroid cells by Dubrovin-Natanzon Cartier divisors on maximal Mumford curves

Abstract

We prove that each totally positive Schubert cell is a real component of Jacobian of the maximal Mumford ($\mathtt{MM}$) curve corresponding to the Le-graph of such cell. At this aim we construct a bijection between the points of this positroid cell and Dubrovin-Natanzon Cartier divisors on the $\mathtt{MM}$-curve.

Version 1: 35 pages, 25 figures. Version 2 (42 pages, 44 figures): fully revised version with addition of new results (Theorem 1 (item 2), Theorems 2,3, and 4). Version 3 (43 pages, 44 figures): references added, minor changes