{"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/monodromy-of-the-family-of-cubic-surfaces","title":"Monodromy of the family of cubic surfaces branching over smooth cubic curves","arxiv_id":"1910.09593","date":"2019-10-21","proceeding":null,"authors":["Adán Medrano Martín del Campo"],"abstract":"Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of $\\mathbb{P}^{2}$, with branch locus equal to a smooth cubic curve. This family is parametrized by the space $\\mathcal{U}_{3}$ of smooth cubic curves in $\\mathbb{P}^{2}$ and each surface is equipped with a $\\mathbb{Z}/3\\mathbb{Z}$ deck group action. We compute the image of the monodromy map $\\rho$ induced by the action of $\\pi_{1}\\left(\\mathcal{U}_{3}\\right)$ on the $27$ lines contained on the cubic surfaces of this family. Due to a classical result, this image is contained in the Weyl group $W\\left(E_{6}\\right)$. Our main result is that $\\rho$ is surjective onto the centralizer of the image a of a generator of the deck group. Our proof is mainly computational, and relies on the relation between the $9$ inflection points in a cubic curve and the $27$ lines contained in the cubic surface branching over it.","url_abs":"https://arxiv.org/abs/1910.09593v2","url_pdf":"https://arxiv.org/pdf/1910.09593v2.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":"monodromy-of-the-family-of-cubic-surfaces","repo_url":"https://github.com/amedranomdelc/Monodromy-of-the-Family-of-Cubic-Surfaces-Branching-over-Smooth-Cubic-Curves","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}