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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2312.16974 |
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| _version_ | 1866914972004515840 |
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| author | Lam, Yeuk Hay Joshua Moretti, Federico Passeri, Giovanni |
| author_facet | Lam, Yeuk Hay Joshua Moretti, Federico Passeri, Giovanni |
| contents | Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications are presented. On the one hand, we show that for a very general curve $C$ and a very general hypersurface $Y\subset \mathbb P^{n+1}$ of degree $\ge 2n+1$, any map $C \to Y$ is constant. On the other hand, we give a lower bound on the genus of a family of curves with an isotrivial factor in the associated family of Jacobians; we also characterize the families of curves attaining this bound as the families of degree $2$ branched covers of a fixed curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16974 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the locus of curves mapping to a fixed target Lam, Yeuk Hay Joshua Moretti, Federico Passeri, Giovanni Algebraic Geometry 14D07, 14J70 Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications are presented. On the one hand, we show that for a very general curve $C$ and a very general hypersurface $Y\subset \mathbb P^{n+1}$ of degree $\ge 2n+1$, any map $C \to Y$ is constant. On the other hand, we give a lower bound on the genus of a family of curves with an isotrivial factor in the associated family of Jacobians; we also characterize the families of curves attaining this bound as the families of degree $2$ branched covers of a fixed curve. |
| title | On the locus of curves mapping to a fixed target |
| topic | Algebraic Geometry 14D07, 14J70 |
| url | https://arxiv.org/abs/2312.16974 |