# An embedding problem for finite local torsors over twisted curves

## TL;DR

This paper investigates a refined embedding problem for finite local torsors over twisted curves in positive characteristic, extending previous work on purely inseparable analogues of the Abhyankar conjecture and confirming the conjecture in the solvable case.

## Contribution

It formulates a refined version of the embedding problem using root stacks and proves its validity for solvable group schemes.

## Key findings

- The refined embedding problem is true for solvable finite local group schemes.
- The approach uses recent developments on tamely ramified torsors and root stacks.
- Partial answers are provided for the purely inseparable analogue of the Abhyankar conjecture.

## Abstract

In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas-Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case.