Semi-log canonical vs $F$-pure singularities

Abstract

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 $\varphi :F_{\ast }{\mathcal{O}}_X\to {\mathcal{O}}_X$, it always extends to a map $\bar{\varphi }$ on the normalization of $X$. In this paper, we study when the surjectivity of $\bar{\varphi }$ implies the surjectivity of $\varphi$. 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.

Minor changes, 21 pages, to appear in the Journal of Algebra