Splitting curves on a rational ruled surface, the Mordell-Weil groups of hyperelliptic fibrations and Zariski pairs

2009/05/01 by Hiro‐o Tokunaga, Hiro-o Tokunaga, Tokunaga, Hiro-o
Mathematics · #14H30 #14J26 #14J27 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:14H30 #msc:14J26 #msc:14J27

paper · pdf · doi:10.48550/arxiv.0905.0047

23pages

arxiv created 2009/05/01 · openalex publication_date 2009/05/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Σ be a smooth projective surface, let f' : S' → Σ be a double cover of Σ and let μ: S → S' be the canonical resolution. Put f = f'∘μ. An irreducible curve C on Σ is said to be a splitting curve with respect to f if f^*C is of the form C⁺ + C^- + E, where C^- = σ_f^*C⁺, σ_f being the covering transformation of f and all irreducible components of E are contained in the exceptional set of μ. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consider the topology of the complements of certain curves on rational ruled surfaces.

Citations

• Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics

Related

• Cyclic coverings of rational normal surfaces which are quotients of a product of curves
• Hyperelliptic d-osculating covers and rational surfaces
• Arrangements of rational sections over curves and the varieties they define
• Rational curves in the moduli of supersingular K3 surfaces
• Separating semigroup of hyperelliptic curves and of genus 3 curves