# On a purely inseparable analogue of the Abhyankar Conjecture for affine curves

## TL;DR

This paper explores a purely inseparable analogue of the Abhyankar Conjecture for affine curves, using Nori's fundamental group scheme, and provides partial results in this area.

## Contribution

It introduces a new analogue of the Abhyankar Conjecture in the context of purely inseparable covers and offers partial solutions using Nori's fundamental group scheme.

## Key findings

- Partial answers to the purely inseparable analogue of the Abhyankar Conjecture.
- Extension of fundamental group concepts to purely inseparable covers.
- Insights into the structure of Nori's fundamental group scheme for affine curves.

## Abstract

Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's \'etale fundamental group of $U$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme, and give a partial answer to it.