A Combinatorial Description of the Dormant Miura Transformation

The aim of the present paper is to describe Miura sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {s}}{\mathfrak {l}}_2$$\end{document}-opers and Miura transformations in terms of graph-theoretic objects. We construct a bijective correspondence between dormant generic Miura sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {s}}{\mathfrak {l}}_2$$\end{document}-opers on a totally degenerate curve in positive characteristic and certain branch numberings on a 3-regular graph. This correspondence allows us to completely identify dormant generic Miura sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {s}}{\mathfrak {l}}_2$$\end{document}-opers on totally degenerate curves. Also, we investigate how this result can be related to the combinatorial description of dormant sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {s}}{\mathfrak {l}}_2$$\end{document}-opers given by S. Mochizuki, F. Liu, and B. Osserman.