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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.15301 |
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| _version_ | 1866908097578008576 |
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| author | Khurmi, Aditya |
| author_facet | Khurmi, Aditya |
| contents | An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an irreducible curve contained in the smooth locus of $X,$ with $p_a(C)=1$ and $C^2=0.$ These naturally arise as blowups $X:=\text{Bl}_e(\mathbb{P}_Δ)$ of projective toric surfaces, whose Newton polygon is elliptic. The order of $\mathcal{O}(C)|_C$ in $\text{Pic}^0(C)$ gives a quantitative way to check if $X$ is an elliptic fibration, which is equivalent to finiteness of the order. We call $Δ$ a Lang-Trotter polygon when this order is infinite, in which case $\overline{\text{Eff}(\text{Bl}_e(\mathbb{P}_Δ))}$ is non-polyhedral. The paper \cite{lizzie} shows there are exactly $3$ elliptic triangles up to $\text{SL}_2(\mathbb{Z}),$ none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when $ρ(\mathbb{P}_Δ)>2.$ Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of $K_X+C.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric Elliptic Pairs with Picard Number Three Khurmi, Aditya Algebraic Geometry An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an irreducible curve contained in the smooth locus of $X,$ with $p_a(C)=1$ and $C^2=0.$ These naturally arise as blowups $X:=\text{Bl}_e(\mathbb{P}_Δ)$ of projective toric surfaces, whose Newton polygon is elliptic. The order of $\mathcal{O}(C)|_C$ in $\text{Pic}^0(C)$ gives a quantitative way to check if $X$ is an elliptic fibration, which is equivalent to finiteness of the order. We call $Δ$ a Lang-Trotter polygon when this order is infinite, in which case $\overline{\text{Eff}(\text{Bl}_e(\mathbb{P}_Δ))}$ is non-polyhedral. The paper \cite{lizzie} shows there are exactly $3$ elliptic triangles up to $\text{SL}_2(\mathbb{Z}),$ none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when $ρ(\mathbb{P}_Δ)>2.$ Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of $K_X+C.$ |
| title | Toric Elliptic Pairs with Picard Number Three |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.15301 |