Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918198524248064 |
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| author | Pérez, Pedro D. González |
| author_facet | Pérez, Pedro D. González |
| contents | Let $(S, 0) \subset (\mathbb{C}^{d+1},0)$ be an irreducible germ of hypersurface. The germ $(S,0)$ is quasi-ordinary if $(S,0)$ has a finite projection to $(\mathbb{C}^d,0)$ which is unramified outside the coordinate hyperplanes. This implies that the normalization of $S$ is a toric singularity. One has also a monomial variety associated to $S$, which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since $(S,0)$ is quasi-ordinary, the extension of the Jacobian ideal of $S$ to the local ring of its normalization is a monomial ideal. We describe this monomial ideal by comparing it with the {\em logarithmic Jacobian ideals} of $S$ and of its associated monomial variety and we give some applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity Pérez, Pedro D. González Algebraic Geometry Primary 14M25, Secondary 32S25 Let $(S, 0) \subset (\mathbb{C}^{d+1},0)$ be an irreducible germ of hypersurface. The germ $(S,0)$ is quasi-ordinary if $(S,0)$ has a finite projection to $(\mathbb{C}^d,0)$ which is unramified outside the coordinate hyperplanes. This implies that the normalization of $S$ is a toric singularity. One has also a monomial variety associated to $S$, which is a toric singularity with the same normalization, and with possibly higher embedding dimension. Since $(S,0)$ is quasi-ordinary, the extension of the Jacobian ideal of $S$ to the local ring of its normalization is a monomial ideal. We describe this monomial ideal by comparing it with the {\em logarithmic Jacobian ideals} of $S$ and of its associated monomial variety and we give some applications. |
| title | Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity |
| topic | Algebraic Geometry Primary 14M25, Secondary 32S25 |
| url | https://arxiv.org/abs/2511.09126 |