On a purely inseparable analogue of the Abhyankar conjecture for affine curves

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<jats:p>Let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X18007194_inline1" /><jats:tex-math>$U$</jats:tex-math></jats:alternatives></jats:inline-formula>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 étale fundamental group<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X18007194_inline2" /><jats:tex-math>$\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U)$</jats:tex-math></jats:alternatives></jats:inline-formula>. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X18007194_inline3" /><jats:tex-math>$\unicode[STIX]{x1D70B}^{N}(U)$</jats:tex-math></jats:alternatives></jats:inline-formula>, and give a partial answer to it.</jats:p>

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