{"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/the-geometry-of-rank-drop-in-a-class-of-face","title":"The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products","arxiv_id":"2301.09826","date":"2023-01-24","proceeding":null,"authors":["Erin Connelly","Sameer Agarwal","Alperen Ergur","Rekha R. Thomas"],"abstract":"Given $k \\leq 6$ points $(x_i,y_i) \\in \\mathbb{P}^2 \\times \\mathbb{P}^2$, we characterize rank deficiency of the $k \\times 9$ matrix $Z_k$ with rows $x_i^\\top \\otimes y_i^\\top$ in terms of the geometry of the point configurations $\\{x_i\\}$ and $\\{y_i\\}$. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For $k \\leq 5$, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case $k=6$ the rank-drop locus is captured by the classical theory of cubic surfaces.","url_abs":"https://arxiv.org/abs/2301.09826v2","url_pdf":"https://arxiv.org/pdf/2301.09826v2.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":"the-geometry-of-rank-drop-in-a-class-of-face","repo_url":"https://github.com/rekharthomas/rankdrop","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}