Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918471287177216 |
|---|---|
| author | Benozzo, Marta Jagathese, Vignesh Pandey, Vaibhav Ramírez-Moreno, Pedro Schwede, Karl Sridhar, Prashanth |
| author_facet | Benozzo, Marta Jagathese, Vignesh Pandey, Vaibhav Ramírez-Moreno, Pedro Schwede, Karl Sridhar, Prashanth |
| contents | We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings Benozzo, Marta Jagathese, Vignesh Pandey, Vaibhav Ramírez-Moreno, Pedro Schwede, Karl Sridhar, Prashanth Commutative Algebra Algebraic Geometry 13A35, 13B22, 14B05, 14F18 We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture. |
| title | Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings |
| topic | Commutative Algebra Algebraic Geometry 13A35, 13B22, 14B05, 14F18 |
| url | https://arxiv.org/abs/2509.07217 |