A note on Severi varieties of nodal curves on Enriques surfaces
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866914694839664640 |
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| author | Ciliberto, C. Dedieu, T. Galati, C. Knutsen, A. L. |
| author_facet | Ciliberto, C. Dedieu, T. Galati, C. Knutsen, A. L. |
| contents | Let $|L|$ be a linear system on a smooth complex Enriques surface $S$ whose general member is a smooth and irreducible curve of genus $p$, with $L^ 2>0$, and let $V_{|L|, δ} (S)$ be the Severi variety of irreducible $δ$-nodal curves in $|L|$. We denote by $π:X\to S$ the universal covering of $S$. In this note we compute the dimensions of the irreducible components $V$ of $V_{|L|, δ} (S)$. In particular we prove that, if $C$ is the curve corresponding to a general element $[C]$ of $V$, then the codimension of $V$ in $|L|$ is $δ$ if $π^{-1}(C)$ is irreducible in $X$ and it is $δ-1$ if $π^ {-1}(C)$ consists of two irreducible components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_06435 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A note on Severi varieties of nodal curves on Enriques surfaces Ciliberto, C. Dedieu, T. Galati, C. Knutsen, A. L. Algebraic Geometry Let $|L|$ be a linear system on a smooth complex Enriques surface $S$ whose general member is a smooth and irreducible curve of genus $p$, with $L^ 2>0$, and let $V_{|L|, δ} (S)$ be the Severi variety of irreducible $δ$-nodal curves in $|L|$. We denote by $π:X\to S$ the universal covering of $S$. In this note we compute the dimensions of the irreducible components $V$ of $V_{|L|, δ} (S)$. In particular we prove that, if $C$ is the curve corresponding to a general element $[C]$ of $V$, then the codimension of $V$ in $|L|$ is $δ$ if $π^{-1}(C)$ is irreducible in $X$ and it is $δ-1$ if $π^ {-1}(C)$ consists of two irreducible components. |
| title | A note on Severi varieties of nodal curves on Enriques surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1811.06435 |