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This paper studies schemes X defined over a field of characteristic p>0 whichadmit a nontrivial $\alpha_p$ or $\mu_p$ action. In particular, the structureof the quotient map $X \rightarrow Y$ is investigated. Information on localproperties of the quotient Y, as singularities and local Picard groups,structure theorems for the quotient map and adjunction formulas for thequotient map are obtained.

Quotients of schemes by $\alpha_p$ or $\mu_p$ actions in characteristic p>0