Explicit Constructions of Halphen Pencils

2020/08/18 by Aline Zanardini, Zanardini, Aline · 1 citation
Computer Science · Mathematics · #14E99 (Secondary) #14J27(Primary) 14H50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2008.08128

openalex publication_date 2020/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct rational elliptic surfaces of index two by explicitly constructing their associated Halphen pencils in the projective plane ℙ². For each of the types of singular fibers that occur we construct at least one example having that type of fiber and in fact, for some, we construct all possible examples. We establish a precise dictionary between the fibers in a rational elliptic surface and the corresponding plane curves and, in particular, we study the singularities of the curves appearing in a Halphen pencil.

Cited by

• Stability of pencils of plane curves, log canonical thresholds and multiplicities

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