Semi-log canonical vs $F$-pure singularities

Details Semi-log canonical vs $F$-pure singularities Semi-log canonical vs $F$-pure singularities

2011-01-05

arxiv.org/abs/1101.1033v3

Journal of Algebra, Vol. 349, Issue 1, 150--164, Jan. 2012

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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