A higher-dimensional Van den Essen type formula for projective foliations and applications
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913654856744960 |
|---|---|
| author | Corrêa, Maurício Costa, Gilcione Nonato |
| author_facet | Corrêa, Maurício Costa, Gilcione Nonato |
| contents | Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ μ(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03947 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A higher-dimensional Van den Essen type formula for projective foliations and applications Corrêa, Maurício Costa, Gilcione Nonato Algebraic Geometry Complex Variables Dynamical Systems Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ μ(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve. |
| title | A higher-dimensional Van den Essen type formula for projective foliations and applications |
| topic | Algebraic Geometry Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2410.03947 |