On weighted bounded negativity for rational surfaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915599098052608 |
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| author | Galindo, Carlos Monserrat, Francisco Moreno-Ávila, Carlos-Jesús |
| author_facet | Galindo, Carlos Monserrat, Francisco Moreno-Ávila, Carlos-Jesús |
| contents | The weighted bounded negativity conjecture considers a smooth projective surface $X$ and looks for a common lower bound on the quotients $C^2/(D\cdot C)^2$, where $C$ runs over the integral curves on $X$ and $D$ over the big and nef divisors on $X$ such that $D \cdot C >0$. We focus our study on rational surfaces $Z$. Setting $π: Z \rightarrow Z_0$ a composition of blowups giving rise to $Z$, where $Z_0$ is the projective plane or a Hirzebruch surface, we give a common lower bound on $C^2/(H^* \cdot C)^2$ whenever $H^*$ is the pull-back of a nef divisor $H$ on $Z_0$. In addition, we prove that, only in the case when a nef divisor $D$ on $Z$ approaches the boundary of the nef cone, the quotients $C^2/(D\cdot C)^2$ could tend to minus infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weighted bounded negativity for rational surfaces Galindo, Carlos Monserrat, Francisco Moreno-Ávila, Carlos-Jesús Algebraic Geometry 14C20, 14E15, 14J26 The weighted bounded negativity conjecture considers a smooth projective surface $X$ and looks for a common lower bound on the quotients $C^2/(D\cdot C)^2$, where $C$ runs over the integral curves on $X$ and $D$ over the big and nef divisors on $X$ such that $D \cdot C >0$. We focus our study on rational surfaces $Z$. Setting $π: Z \rightarrow Z_0$ a composition of blowups giving rise to $Z$, where $Z_0$ is the projective plane or a Hirzebruch surface, we give a common lower bound on $C^2/(H^* \cdot C)^2$ whenever $H^*$ is the pull-back of a nef divisor $H$ on $Z_0$. In addition, we prove that, only in the case when a nef divisor $D$ on $Z$ approaches the boundary of the nef cone, the quotients $C^2/(D\cdot C)^2$ could tend to minus infinity. |
| title | On weighted bounded negativity for rational surfaces |
| topic | Algebraic Geometry 14C20, 14E15, 14J26 |
| url | https://arxiv.org/abs/2408.05466 |