2020/08/07 by Noriyuki Hamada, Hamada, Noriyuki, Kenta Hayano +1 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2008.02925
openalex publication_date 2020/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we classify relatively minimal genus-1 holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on ℙ1× ℙ1 of bi-degree (2,2) or a blow-up of the pencil on ℙ2 of degree 3, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-1 Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on ℙ2 of degree 3 does not depend on the choice of blown-up base points. We also show that the genus-1 Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on ℙ2 and ℙ1× ℙ1 above, in particular these are both holomorphic.