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 $\phi : F_* \mathcal{ O}_X \to \mathcal{ O}_X$, it always extends to a map $\bar{ \phi}$ on the normalization of $X$. In this paper, we study when the surjectivity of $\bar{ \phi}$ implies the surjectivity of $\phi$. 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.
Categories: math.AG math.AC