Moduli space of genus one curves on cubic threefold
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914546784927744 |
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| author | Feng, Enhao |
| author_facet | Feng, Enhao |
| contents | Let $X$ be a smooth cubic threefold. By invoking ideas from Geometric Manin's Conjecture, we give a complete description of the main components of the Kontsevich moduli space of genus one stable maps $\overline{M}_{1,0}(X)$. In particular, we show that for degree $e\geqslant 5$, there are exactly two irreducible main components, of which one generically parametrizes free curves birational onto their images, and the other corresponds to degree $e$ covers of lines. As a corollary, we classify components of the morphism space $\text{Mor}(E,X)$ for a general smooth genus one curve $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16686 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moduli space of genus one curves on cubic threefold Feng, Enhao Algebraic Geometry Let $X$ be a smooth cubic threefold. By invoking ideas from Geometric Manin's Conjecture, we give a complete description of the main components of the Kontsevich moduli space of genus one stable maps $\overline{M}_{1,0}(X)$. In particular, we show that for degree $e\geqslant 5$, there are exactly two irreducible main components, of which one generically parametrizes free curves birational onto their images, and the other corresponds to degree $e$ covers of lines. As a corollary, we classify components of the morphism space $\text{Mor}(E,X)$ for a general smooth genus one curve $E$. |
| title | Moduli space of genus one curves on cubic threefold |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.16686 |