Polarized K3 surfaces with an automorphism of order 3 and low Picard number
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916173888618496 |
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| author | Festi, Dino |
| author_facet | Festi, Dino |
| contents | In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $ρ(X):=\textrm{rank } \textrm{Pic } X = h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in\{ 4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer $h^*_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h^*_{3,2d}$ is equal to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17539 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polarized K3 surfaces with an automorphism of order 3 and low Picard number Festi, Dino Algebraic Geometry 14J28, 14J50, 14J10 In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $ρ(X):=\textrm{rank } \textrm{Pic } X = h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in\{ 4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer $h^*_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h^*_{3,2d}$ is equal to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds. |
| title | Polarized K3 surfaces with an automorphism of order 3 and low Picard number |
| topic | Algebraic Geometry 14J28, 14J50, 14J10 |
| url | https://arxiv.org/abs/2306.17539 |