Semi-log canonical vs $F$-pure singularities

If $X$ is Frobenius split, then so is its normalization and we explore conditions which imply the converse. To do this, we recall that given an $\mathcal{O}_X$-linear map $ϕ: F_* \mathcal{O}_X \to \mathcal{O}_X$, it always extends to a map $\barϕ$ on the normalization of $X$. In this paper, we study when the surjectivity of $\barϕ$ implies the surjectivity of $ϕ$. While this doesn't occur generally, we show it always happens if certain tameness conditions are satisfied for the normalization map. Our result has geometric consequences including a connection between $F$-pure singularities and semi-log canonical singularities, and a more familiar version of the ($F$-)inversion of adjunction formula.

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