2023/01/12 by Ananyo Dan, Dan, Ananyo
Computer Science · Engineering · Mathematics · #14C30 #32S35 #32S50 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2301.04940
openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we give a counterexample to the Lefschetz hyperplane theorem for non-singular quasi-projective varieties. A classical result of Hamm-Lê shows that Lefschetz hyperplane theorem can hold for hyperplanes in general position. We observe that the condition of ``hyperplane'' is strict in the sense that it is not possible to replace it by higher degree hypersurfaces. The counterexample is very simple: projective space minus finitely many points. Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails in the quasi-projective case.