On Stability and Denominators of F-pure thresholds in Families of Diagonal Hypersurfaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917649648189440 |
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| author | Chua, Gari Lincoln |
| author_facet | Chua, Gari Lincoln |
| contents | Given a prime number $p$ and a positive integer $m$, we provide a family of diagonal hypersurfaces $\{ f_n \}_{n = 1}^{\infty}$ in $m$ variables, for which the denominator of $\text{ fpt } (f_{n})$ (in lowest terms) is always $p$ and whose $F$-pure thresholds stabilize after a certain $n$. We also provide another family of diagonal hypersurfaces $\{ g_n \}_{n = 1}^{\infty}$ in $m$ variables, for which the power of $p$ in the denominator of $\text{ fpt } (g_{n})$ (in lowest terms) diverges to $\infty$ as $n \to \infty$. This behavior of the denominator of the $F$-pure thresholds is dependent on the congruence class of $p$ modulo the smallest two exponents of $\{ f_n \}$ and $\{ g_n \}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Stability and Denominators of F-pure thresholds in Families of Diagonal Hypersurfaces Chua, Gari Lincoln Commutative Algebra Algebraic Geometry 13A35, 14B05, 13P99 Given a prime number $p$ and a positive integer $m$, we provide a family of diagonal hypersurfaces $\{ f_n \}_{n = 1}^{\infty}$ in $m$ variables, for which the denominator of $\text{ fpt } (f_{n})$ (in lowest terms) is always $p$ and whose $F$-pure thresholds stabilize after a certain $n$. We also provide another family of diagonal hypersurfaces $\{ g_n \}_{n = 1}^{\infty}$ in $m$ variables, for which the power of $p$ in the denominator of $\text{ fpt } (g_{n})$ (in lowest terms) diverges to $\infty$ as $n \to \infty$. This behavior of the denominator of the $F$-pure thresholds is dependent on the congruence class of $p$ modulo the smallest two exponents of $\{ f_n \}$ and $\{ g_n \}$. |
| title | On Stability and Denominators of F-pure thresholds in Families of Diagonal Hypersurfaces |
| topic | Commutative Algebra Algebraic Geometry 13A35, 14B05, 13P99 |
| url | https://arxiv.org/abs/2404.10968 |