Shafarevich's conjecture for families of hypersurfaces over function fields
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916830550949888 |
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| author | Engel, Philip Lin, Alice Tayou, Salim |
| author_facet | Engel, Philip Lin, Alice Tayou, Salim |
| contents | Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}_{\mathbb C}^{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}_{\mathbb C}^{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shafarevich's conjecture for families of hypersurfaces over function fields Engel, Philip Lin, Alice Tayou, Salim Algebraic Geometry Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}_{\mathbb C}^{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}_{\mathbb C}^{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. |
| title | Shafarevich's conjecture for families of hypersurfaces over function fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.06387 |