Semi-log canonical vs $F$-pure singularities

TL;DR

This paper investigates the relationship between Frobenius split and $F$-pure singularities, establishing conditions under which properties of the normalization imply properties of the original variety, with implications for singularity classification.

Contribution

It provides new criteria linking $F$-purity and semi-log canonical singularities via tameness conditions on the normalization map.

Findings

Surjectivity of extended maps implies original map surjectivity under tameness.

Connection established between $F$-pure and semi-log canonical singularities.

Results include a version of the ($F$-)inversion of adjunction formula.

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 $\phi : F_* \mathcal{O}_X \to \mathcal{O}_X$, 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.