Free curves, Eigenschemes and Pencils of curves
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917967360425984 |
|---|---|
| author | Di Gennaro, Roberta Ilardi, Giovanna Mirò-Roig, Rosa Maria Schenck, Hal Vallès, Jean |
| author_facet | Di Gennaro, Roberta Ilardi, Giovanna Mirò-Roig, Rosa Maria Schenck, Hal Vallès, Jean |
| contents | Let $R=K[x,y,z]$. A reduced plane curve $C=V(f)\subset \mathbf P^2$ is $free \ $ if its associated module of tangent derivations $\mathrm{Der}(f)$ is a free $R$-module, or equivalently if the corresponding sheaf $T_ {\mathbf P^2 }(-\log C)$ of vector fields tangent to $C$ splits as a direct sum of line bundles on $\mathbf P^2$. In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in $\mathbf P^2$. The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09443 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Free curves, Eigenschemes and Pencils of curves Di Gennaro, Roberta Ilardi, Giovanna Mirò-Roig, Rosa Maria Schenck, Hal Vallès, Jean Algebraic Geometry Commutative Algebra Primary 14H20, Secondary 14C21, 14J60 Let $R=K[x,y,z]$. A reduced plane curve $C=V(f)\subset \mathbf P^2$ is $free \ $ if its associated module of tangent derivations $\mathrm{Der}(f)$ is a free $R$-module, or equivalently if the corresponding sheaf $T_ {\mathbf P^2 }(-\log C)$ of vector fields tangent to $C$ splits as a direct sum of line bundles on $\mathbf P^2$. In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in $\mathbf P^2$. The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves. |
| title | Free curves, Eigenschemes and Pencils of curves |
| topic | Algebraic Geometry Commutative Algebra Primary 14H20, Secondary 14C21, 14J60 |
| url | https://arxiv.org/abs/2306.09443 |