The defect of the F-pure threshold
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910071261233152 |
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| author | De Stefani, Alessandro Núñez-Betancourt, Luis Smirnov, Ilya |
| author_facet | De Stefani, Alessandro Núñez-Betancourt, Luis Smirnov, Ilya |
| contents | Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The defect of the F-pure threshold De Stefani, Alessandro Núñez-Betancourt, Luis Smirnov, Ilya Commutative Algebra Algebraic Geometry Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring. |
| title | The defect of the F-pure threshold |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2501.13613 |