Asymptotic Betti bounds for hypersurfaces in a singular variety
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914285787021312 |
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| author | Pan, Xuanyu Zhang, Dingxin Zhang, Xiping |
| author_facet | Pan, Xuanyu Zhang, Dingxin Zhang, Xiping |
| contents | We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant $C > 0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20179 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Betti bounds for hypersurfaces in a singular variety Pan, Xuanyu Zhang, Dingxin Zhang, Xiping Algebraic Geometry 14F45, 14C17, 14F20 We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant $C > 0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves. |
| title | Asymptotic Betti bounds for hypersurfaces in a singular variety |
| topic | Algebraic Geometry 14F45, 14C17, 14F20 |
| url | https://arxiv.org/abs/2601.20179 |