{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/k3-surfaces-with-two-involutions-and-low","title":"K3 surfaces with two involutions and low Picard number","arxiv_id":"2210.14623","date":"2022-10-26","proceeding":null,"authors":["Dino Festi","Wim Nijgh","Daniel Platt"],"abstract":"Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $\\rho$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $\\rho \\geq 1$ when $d=1$ and $\\rho \\geq 2$ when $d \\geq 2$. For $d=1$, the first example defined over $\\mathbb{Q}$ with $\\rho=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\\={o}, also defined over $\\mathbb{Q}$, can be used to realise the minimum $\\rho=2$ for all $d\\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $\\rho=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $\\rho=2$ 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 $\\textrm{Pic} Y_\\mathbb{C}=\\textrm{Pic} Y \\cong N$.","url_abs":"https://arxiv.org/abs/2210.14623v2","url_pdf":"https://arxiv.org/pdf/2210.14623v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"k3-surfaces-with-two-involutions-and-low","repo_url":"https://github.com/danielplatt/quartic-k3-with-involution","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}