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

Hiro‐o Tokunaga|ArXiv.org|May 1, 2009

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper establishes a reciprocity law for splitting curves on rational ruled surfaces, linking the topology of double covers and Mordell-Weil groups of hyperelliptic fibrations. It proves that the splitting behavior of curves under double covers is governed by the arithmetic of sections on elliptically fibered surfaces, leading to new constructions of Zariski pairs of sextic curves via geometric and arithmetic duality.

ABSTRACT

Let $Σ$ be a smooth projective surface, let $f' : S' o Σ$ be a double cover of $Σ$ and let $μ: S o S'$ be the canonical resolution. Put $f = f'\circμ$. 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.

Motivation & Objective

• To formulate a geometric reciprocity law for splitting curves on rational ruled surfaces.
• To study the topology of complements of curves on rational ruled surfaces via double covers and elliptic fibrations.
• To construct new Zariski pairs of sextic curves using the interplay between Mordell-Weil groups and splitting divisors.
• To provide a geometric interpretation of quadratic reciprocity in the context of algebraic surfaces and branched covers.

Proposed method

• Define splitting curves as irreducible curves whose pullback under a double cover decomposes into conjugate components plus exceptional divisors.
• Use canonical resolutions of double covers to analyze the structure of pullbacks and their components.
• Relate the splitting behavior of curves to the Mordell-Weil group of the associated hyperelliptic fibration on a rational ruled surface.
• Construct double covers of Hirzebruch surfaces with specified branch loci, using divisors of the form $\Delta_{0,2n} + T_{2n}^{(\nu)}$ with $T_{2n}^{(\nu)}$ irreducible and with simple singularities.
• Leverage the group law on generic fibers of elliptic fibrations to identify sections arising from torsion points and relate them to conic and quartic curves.
• Apply the criterion from [2] to determine when a $D_{2n}$-cover branched at $2C + nQ$ exists, based on splitting conditions and divisibility in the Mordell-Weil group.

Experimental results

Research questions

• RQ1 Under what conditions does a curve $D$ on a rational ruled surface become a splitting curve with respect to a double cover?
• RQ2 How does the Mordell-Weil group of a hyperelliptic fibration on a rational ruled surface control the splitting behavior of curves?
• RQ3 Can the reciprocity law for splitting curves be used to construct Zariski pairs of sextic curves?
• RQ4 What is the geometric and arithmetic significance of the splitting condition $f^*\mathcal{D} = \mathcal{D}^+ + \mathcal{D}^- + E$ in the context of double covers?

Key findings

• A reciprocity law for splitting curves holds on rational ruled surfaces, where the splitting of one curve under a double cover is determined by the Mordell-Weil group of the associated fibration.
• The existence of a $D_{2n}$-cover branched at $2C + nQ$ is equivalent to the existence of a $D_{2n}$-cover branched at $2(\Delta_{0,2} + \Delta_C) + nT_Q$ for a quartic $Q$ with $2a_1$ or $a_3$ singularities.
• Two Zariski pairs of sextic curves are constructed: one from a quartic with $2a_1$ singularities, and another from a quartic with one $a_3$ singularity, both tangent to conics at four smooth points.
• The sections $2s_o$ and $s_1 + s_2$ on the elliptic surface correspond to conics $C_1$ and $C_2$, respectively, and their distinct topology leads to non-homeomorphic pairs $(C_1 \cup Q, C_2 \cup Q)$.
• The construction confirms that the pairs are Zariski pairs, and these examples are included in Shimada’s list, but the paper provides a new geometric-arithmetic explanation via Mordell-Weil groups.
• The Mordell-Weil group of the resolved double cover has no 2-torsion, implying that the branch divisor $T_{2n}^{(\nu_n)}$ is irreducible, which is essential for the construction.