A Criterion for Perfectoid Purity and the Rationality of Thresholds

Abstract

We introduce a new criterion providing a sufficient condition for a hypersurface in an unramified regular local ring to be perfectoid pure. The criterion is formulated in terms of an explicitly computable sequence of integers, called the splitting-order sequence. Our main theorem shows that if all entries of the sequence are at most $p-1$, then the hypersurface is perfectoid pure, and the perfectoid-pure threshold can be computed explicitly from it. As a consequence, we prove that for any regular local ring $R$, the perfectoid pure threshold $\mathrm{ppt}(R,p)$ with respect to $p$ is always a rational number. Moreover, we show that for sufficiently large primes $p$, the cone over a Fermat type Calabi-Yau hypersurface is perfectoid pure, revealing new and unexpected examples of perfectoid pure singularities. Moreover, we show that for sufficiently large primes $p$, the cone over a Fermat type Calabi-Yau hypersurface is perfectoid pure, revealing new and unexpected examples of perfectoid pure singularities.