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 OX-linear map ϕ:F⁎OX→OX, it always extends to a map on the normalization of X. In this paper, we study when the surjectivity of implies the surjectivity of ϕ. While this doesnʼt 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.