Arithmetic and Geometric Deformations of F -Pure and F -Regular Singularities

abstract: Given a flat family $\mathcal{X}$ of $\Q$-Gorenstein singularities over an open subset $U$ of $\Spec\Z$, our first main result says that if the total space $\mathcal{X}$ is $\Q$-Gorenstein and a mod-$p$ fiber is $F$-pure for some $p \in U$, then the generic fiber is log canonical. We also show the analog of this result for log terminal singularities, without assuming that $\mathcal{X}$ is $\Q$-Gorenstein, as a generalization of the result of Ma-Schwede. Our second main result shows that two-dimensional, strongly $F$-regular singularities are stable under equal characteristic deformations. Our results provide an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto regarding the deformations of linearly reductive quotient singularities.