On the genus of a curve in a projective $3$-fold
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915753508208640 |
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| author | Di Gennaro, Vincenzo Rapagnetta, Antonio Sabatino, Pietro |
| author_facet | Di Gennaro, Vincenzo Rapagnetta, Antonio Sabatino, Pietro |
| contents | Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a projective subscheme of dimension $1$, degree $d\gg n$, and arithmetic genus $p_a(C)$. Improving a recent result by Liu, we exhibit a Castelnuovo's bound for $p_a(C)$. In the case $X$ is Calabi-Yau, our bound gives a step forward for a certain conjecture concerning the vanishing of Gopakumar-Vafa invariants of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17852 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the genus of a curve in a projective $3$-fold Di Gennaro, Vincenzo Rapagnetta, Antonio Sabatino, Pietro Algebraic Geometry Let $X\subset \mathbb P^r$ be a projective factorial variety of dimension $3$, degree $n$, with at worst isolated singularities. Assume that the Picard group of $X$ is generated by the hyperplane section class. Let $C\subset X$ be a projective subscheme of dimension $1$, degree $d\gg n$, and arithmetic genus $p_a(C)$. Improving a recent result by Liu, we exhibit a Castelnuovo's bound for $p_a(C)$. In the case $X$ is Calabi-Yau, our bound gives a step forward for a certain conjecture concerning the vanishing of Gopakumar-Vafa invariants of $X$. |
| title | On the genus of a curve in a projective $3$-fold |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2601.17852 |