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| Format: | Preprint |
| Published: |
2014
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1409.4175 |
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| _version_ | 1866915096248188928 |
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| author | Drézet, Jean-Marc |
| author_facet | Drézet, Jean-Marc |
| contents | A primitive multiple curve is a Cohen-Macaulay irreducible projective curve $Y$ that can be locally embedded in a smooth surface, and such that $C=Y_{red}$ is smooth. In this case, $L={\mathcal I}_C/{\mathcal I}_C^2$ is a line bundle on $C$.
This paper continues the study of deformations of $Y$ to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity $n$ of $Y$). We prove that a primitive double curve can be deformed to reduced curves with smooth components intersecting transversally if and only if $h^0(L^{-1})\not=0$. We give also some properties of reducible deformations in the case of multiplicity $n>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1409_4175 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Reducible deformations and smoothing of primitive multiple curves Drézet, Jean-Marc Algebraic Geometry 14M05, 14B20 A primitive multiple curve is a Cohen-Macaulay irreducible projective curve $Y$ that can be locally embedded in a smooth surface, and such that $C=Y_{red}$ is smooth. In this case, $L={\mathcal I}_C/{\mathcal I}_C^2$ is a line bundle on $C$. This paper continues the study of deformations of $Y$ to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity $n$ of $Y$). We prove that a primitive double curve can be deformed to reduced curves with smooth components intersecting transversally if and only if $h^0(L^{-1})\not=0$. We give also some properties of reducible deformations in the case of multiplicity $n>2$. |
| title | Reducible deformations and smoothing of primitive multiple curves |
| topic | Algebraic Geometry 14M05, 14B20 |
| url | https://arxiv.org/abs/1409.4175 |