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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2305.12555 |
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| _version_ | 1866912746933583872 |
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| author | Cuadrado, Pablo Portilla Sigurðsson, Baldur |
| author_facet | Cuadrado, Pablo Portilla Sigurðsson, Baldur |
| contents | For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field $-\nabla \log |f|$. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12555 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The total spine of the Milnor fibration of a plane curve singularity Cuadrado, Pablo Portilla Sigurðsson, Baldur Algebraic Geometry Dynamical Systems For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field $-\nabla \log |f|$. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary. |
| title | The total spine of the Milnor fibration of a plane curve singularity |
| topic | Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2305.12555 |