# A combinatorial description of the dormant Miura transformation

## TL;DR

This paper provides a combinatorial framework for understanding dormant generic Miura $ ext{sl}_2$-opers on totally degenerate curves, linking algebraic geometry with graph theory to analyze the Miura transformation.

## Contribution

It introduces a novel combinatorial description of dormant Miura $ ext{sl}_2$-opers using branch numberings of 3-regular graphs, extending previous work on dormant $ ext{sl}_2$-opers.

## Key findings

- Combinatorial description of dormant Miura $ ext{sl}_2$-opers on degenerate curves.
- Identification of these opers on curves of genus greater than zero.
- Connection of the Miura transformation to combinatorial objects.

## Abstract

A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat $\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant generic Miura $\mathfrak{sl}_2$-opers on a totally degenerate curve. The combinatorial objects that we use are certain branch numberings of $3$-regular graphs. Our description may be thought of as an analogue of the combinatorial description of dormant $\mathfrak{sl}_2$-opers given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the Miura transformation in terms of combinatorics. As an application, we identify the dormant generic Miura $\mathfrak{sl}_2$-opers on totally degenerate curves of genus $>0$.