Non-free sections of Fano fibrations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912299604770816 |
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| author | Lehmann, Brian Riedl, Eric Tanimoto, Sho |
| author_facet | Lehmann, Brian Riedl, Eric Tanimoto, Sho |
| contents | Let $B$ be a smooth projective curve and let $π: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component $M$ the sections come from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of $\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there is a bounded family of morphisms $f$ which together account for all such components $M$. These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod $p$, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01695 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-free sections of Fano fibrations Lehmann, Brian Riedl, Eric Tanimoto, Sho Algebraic Geometry Number Theory Let $B$ be a smooth projective curve and let $π: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component $M$ the sections come from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of $\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there is a bounded family of morphisms $f$ which together account for all such components $M$. These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod $p$, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height. |
| title | Non-free sections of Fano fibrations |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2301.01695 |