Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908476841656320 |
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| author | Ciliberto, Ciro Miranda, Rick Roé, Joaquim |
| author_facet | Ciliberto, Ciro Miranda, Rick Roé, Joaquim |
| contents | Let $X_n$ be the projective plane blown up at $n \geq 10$ general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on $X_n$. We begin by classifying such systems $|C|$ with general irreducible member of genus $g \geq 2$ (up to Cremona equivalence), in terms of invariants of the adjoint systems $|C+mK|$. We then use this to prove that, for fixed $n \geq 10$ and $g\geq 2$, up to the action of the Cremona group, there exist finitely many complete linear systems on $X_n$ whose general member is irreducible of genus $g$. Further, there is a function $g\mapsto n(g)$ such that every such (effective) system is Cremona equivalent to a system in $X_{n(g)}$. The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given $n$ and $\dim(|C|)$. We classify those systems which achieve the minimal self-intersection. We also classify the systems with $C^2 \leq 5$, whether or not they have minimal $C^2$ for the given $n$ and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture Ciliberto, Ciro Miranda, Rick Roé, Joaquim Algebraic Geometry 14H45, 14H50, 14J26 Let $X_n$ be the projective plane blown up at $n \geq 10$ general points. In this paper we give several consequences of the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, that pertain to complete linear systems on $X_n$. We begin by classifying such systems $|C|$ with general irreducible member of genus $g \geq 2$ (up to Cremona equivalence), in terms of invariants of the adjoint systems $|C+mK|$. We then use this to prove that, for fixed $n \geq 10$ and $g\geq 2$, up to the action of the Cremona group, there exist finitely many complete linear systems on $X_n$ whose general member is irreducible of genus $g$. Further, there is a function $g\mapsto n(g)$ such that every such (effective) system is Cremona equivalent to a system in $X_{n(g)}$. The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given $n$ and $\dim(|C|)$. We classify those systems which achieve the minimal self-intersection. We also classify the systems with $C^2 \leq 5$, whether or not they have minimal $C^2$ for the given $n$ and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image. |
| title | Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture |
| topic | Algebraic Geometry 14H45, 14H50, 14J26 |
| url | https://arxiv.org/abs/2508.01462 |