Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.19319 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910157953302528 |
|---|---|
| author | Yoshikawa, Shou |
| author_facet | Yoshikawa, Shou |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Criterion for Perfectoid Purity and the Rationality of Thresholds Yoshikawa, Shou Algebraic Geometry 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. |
| title | A Criterion for Perfectoid Purity and the Rationality of Thresholds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.19319 |