The multiplicity of cyclic coverings of a singularity of an algebraic variety
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914200788402176 |
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| author | Kumagai, Toya Okuma, Tomohiro |
| author_facet | Kumagai, Toya Okuma, Tomohiro |
| contents | Let $V$ be an affine algebraic variety, and let $p\in V$ be a singular point. For a regular function $g$ on $V$ such that $g(p)=0$ and for a positive integer $n$, we consider the cyclic covering $ϕ_n\: V_n \to V$ of degree $n$ branched along the hypersurface defined by $g$. We will prove that for sufficiently large $n$, the tangent cone of $V_n$ at $ϕ_n^{-1}(p)$ is, as an affine variety, the product of the tangent cone of the branch locus and the affine line. In particular, the multiplicity of the singularity $ϕ_n^{-1}(p) \in V_n$, which is a function of $n$ determined by $V$ and $g$, remains constant for sufficiently large $n$. This result generalizes Tomaru's theorem for normal surface singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14355 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The multiplicity of cyclic coverings of a singularity of an algebraic variety Kumagai, Toya Okuma, Tomohiro Algebraic Geometry Commutative Algebra Primary 14B05, Secondary 13P10, 14J17, 32S05 Let $V$ be an affine algebraic variety, and let $p\in V$ be a singular point. For a regular function $g$ on $V$ such that $g(p)=0$ and for a positive integer $n$, we consider the cyclic covering $ϕ_n\: V_n \to V$ of degree $n$ branched along the hypersurface defined by $g$. We will prove that for sufficiently large $n$, the tangent cone of $V_n$ at $ϕ_n^{-1}(p)$ is, as an affine variety, the product of the tangent cone of the branch locus and the affine line. In particular, the multiplicity of the singularity $ϕ_n^{-1}(p) \in V_n$, which is a function of $n$ determined by $V$ and $g$, remains constant for sufficiently large $n$. This result generalizes Tomaru's theorem for normal surface singularities. |
| title | The multiplicity of cyclic coverings of a singularity of an algebraic variety |
| topic | Algebraic Geometry Commutative Algebra Primary 14B05, Secondary 13P10, 14J17, 32S05 |
| url | https://arxiv.org/abs/2403.14355 |