On regular but non-smooth integral curves
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913494529474560 |
|---|---|
| author | Hilario, Cesar Stöhr, Karl-Otto |
| author_facet | Hilario, Cesar Stöhr, Karl-Otto |
| contents | Let $C$ be a regular geometrically integral curve over an imperfect field $K$ and assume that it admits a non-smooth point $\mathfrak{p}$ which -- seen as a prime of the separable function field $K(C)|K$ -- is non-decomposed in the base field extension $\overline{K} \otimes_K K(C)|\overline{K}$. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform $\mathfrak{p}$ into a rational point. This provides an algorithm to compute geometric $δ$-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed $δ$-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16962 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On regular but non-smooth integral curves Hilario, Cesar Stöhr, Karl-Otto Algebraic Geometry 14G17, 14H05, 14H45, 14D06 Let $C$ be a regular geometrically integral curve over an imperfect field $K$ and assume that it admits a non-smooth point $\mathfrak{p}$ which -- seen as a prime of the separable function field $K(C)|K$ -- is non-decomposed in the base field extension $\overline{K} \otimes_K K(C)|\overline{K}$. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform $\mathfrak{p}$ into a rational point. This provides an algorithm to compute geometric $δ$-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed $δ$-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest. |
| title | On regular but non-smooth integral curves |
| topic | Algebraic Geometry 14G17, 14H05, 14H45, 14D06 |
| url | https://arxiv.org/abs/2211.16962 |