A Sylvester–Gallai theorem for cubic curves. (June 2022)

• Record Type: Journal Article
• Authors: Cohen, Alex; de Zeeuw, Frank
• Is Part Of: European journal of combinatorics. Volume 103(2022)
• Publication Date: 2022-06
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2022.103509
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We prove a variant of the Sylvester–Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in R 2 is not contained in a cubic, then there is a cubic that contains exactly nine of the points. This resolves the first unknown case of a conjecture of Wiseman and Wilson from 1988, who proved a variant of Sylvester–Gallai for conics and conjectured that similar statements hold for curves of any degree.