Rational elliptic surfaces with six singular double fibres
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914324697579520 |
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| author | Ciliberto, Ciro Grassi, Antonella Miranda, Rick Verra, Alessandro Zanardini, Aline |
| author_facet | Ciliberto, Ciro Grassi, Antonella Miranda, Rick Verra, Alessandro Zanardini, Aline |
| contents | A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11725 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational elliptic surfaces with six singular double fibres Ciliberto, Ciro Grassi, Antonella Miranda, Rick Verra, Alessandro Zanardini, Aline Algebraic Geometry A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it. |
| title | Rational elliptic surfaces with six singular double fibres |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2602.11725 |