K3 surfaces with two involutions and low Picard number
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914166705487872 |
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| author | Festi, Dino Nijgh, Wim Platt, Daniel |
| author_facet | Festi, Dino Nijgh, Wim Platt, Daniel |
| contents | Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $ρ$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $ρ\geq 1$ when $d=1$ and $ρ\geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $ρ=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over $\mathbb{Q}$, can be used to realise the minimum $ρ=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $ρ=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $ρ=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14623 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | K3 surfaces with two involutions and low Picard number Festi, Dino Nijgh, Wim Platt, Daniel Algebraic Geometry Differential Geometry Number Theory 14J28, 14C22, 11G99 Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $ρ$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $ρ\geq 1$ when $d=1$ and $ρ\geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $ρ=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over $\mathbb{Q}$, can be used to realise the minimum $ρ=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $ρ=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $ρ=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$. |
| title | K3 surfaces with two involutions and low Picard number |
| topic | Algebraic Geometry Differential Geometry Number Theory 14J28, 14C22, 11G99 |
| url | https://arxiv.org/abs/2210.14623 |