A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918328120901632 |
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| author | Hase-Liu, Matthew |
| author_facet | Hase-Liu, Matthew |
| contents | Geometric Manin's conjecture predicts that components of the moduli space of curves on a Fano variety parametrizing non-free curves are pathological and arise from "accumulating" morphisms that increase the Fujita invariant. By passing to positive characteristic and employing a higher genus generalization of the circle method, we prove a converse to this conjecture for general hypersurfaces $X$ in $\mathbb{P}^{n}$ of degree $d\le n/4+3/2$, namely that there are no such accumulating maps to $X$. One consequence of this is a version of Poincaré duality for these moduli spaces in a range. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality Hase-Liu, Matthew Algebraic Geometry Number Theory Geometric Manin's conjecture predicts that components of the moduli space of curves on a Fano variety parametrizing non-free curves are pathological and arise from "accumulating" morphisms that increase the Fujita invariant. By passing to positive characteristic and employing a higher genus generalization of the circle method, we prove a converse to this conjecture for general hypersurfaces $X$ in $\mathbb{P}^{n}$ of degree $d\le n/4+3/2$, namely that there are no such accumulating maps to $X$. One consequence of this is a version of Poincaré duality for these moduli spaces in a range. |
| title | A converse to geometric Manin's conjecture for general low degree hypersurfaces and Poincaré duality |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2501.12506 |