[Paper Review] Pencils and Infinite Dihedral covers of P^2

Enrique Artal Bartolo, Cogolludo, Jose Ignacio+1 more|arXiv (Cornell University)|Nov 23, 2004

Mathematics and Applications4 citations

TL;DR

This paper establishes a deep connection between the existence of finite dihedral covers of the projective plane branched along a curve $C = 2C_1 + nC_2$, infinite dihedral covers, and the existence of pencils of curves containing $C_1 \cup C_2$. It proves that if such $D_{2n}$-covers exist for infinitely many odd $n$, then they exist for all $n$, and this occurs precisely when $F_2 = G_1^2 - G_2^2 F_1$ for homogeneous polynomials $G_1, G_2$, implying a surjection from the fundamental group of the complement onto the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$. This links topology, algebraic geometry, and Galois covers.

ABSTRACT

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

Motivation & Objective

• To investigate the conditions under which finite dihedral covers $D_{2n}$ of $\mathbb{P}^2$ branched along $2C_1 + nC_2$ exist, where $C_1$ and $C_2$ are irreducible components.
• To relate the existence of such finite dihedral covers to the existence of pencils of curves containing $C_1 \cup C_2$.
• To establish a connection between these covers and the existence of infinite dihedral covers of $\mathbb{P}^2$, particularly via the fundamental group of the complement.
• To derive necessary and sufficient algebraic conditions (in terms of defining polynomials) for the existence of such covers, especially when $n$ varies over odd integers.

Proposed method

• Use of Galois theory and field extensions: a $D_{2n}$-cover corresponds to a Galois extension $\mathbb{C}(X)/\mathbb{C}(Y)$ with Galois group $D_{2n}$, where $Y = \mathbb{P}^2$.
• Analysis of the double cover $\delta: Z \to \mathbb{P}^2$ ramified along $C_1$, and study of the preimage of $C_2$ under this cover, which splits into $C_2^+ \cup C_2^-$.
• Application of topological and algebraic conditions: the condition that $C_2^+ - C_2^-$ is divisible by $n$ in the Picard group is essential for the existence of $D_{2n}$-covers.
• Use of the fundamental group of the complement $\mathbb{P}^2 \setminus (C_1 \cup C_2)$ and its surjection onto $\mathbb{Z}_2 * \mathbb{Z}_2$, derived from the existence of pencils.
• Construction of a pencil via homogeneous polynomials $G_1, G_2$ such that $F_2 = G_1^2 - G_2^2 F_1$, which ensures the existence of the required covers.
• Leveraging results from Zariski and others on fundamental groups of curve complements and their surjections onto free products of cyclic groups.

Experimental results

Research questions

• RQ1 Under what algebraic conditions does a $D_{2n}$-cover of $\mathbb{P}^2$ branched along $2C_1 + nC_2$ exist for infinitely many odd $n$?
• RQ2 How is the existence of such finite dihedral covers related to the existence of a pencil of curves containing $C_1$ and $C_2$?
• RQ3 What is the precise algebraic condition on the defining polynomials $F_1, F_2$ of $C_1$ and $C_2$ that ensures the existence of $D_{2n}$-covers for all $n$?
• RQ4 How does the fundamental group of $\mathbb{P}^2 \setminus (C_1 \cup C_2)$ relate to the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$?
• RQ5 What role does the self-intersection of $C_2^+ - C_2^-$ play in determining the existence of $D_{2n}$-covers?

Key findings

• If $D_{2n}$-covers branched at $2C_1 + nC_2$ exist for infinitely many odd $n$, then they exist for all $n \in \mathbb{N}$, establishing a strong finiteness condition.
• The defining equations $F_1, F_2$ of $C_1$ and $C_2$ must satisfy $F_2 = G_1^2 - G_2^2 F_1$ for some homogeneous polynomials $G_1, G_2$, which is both necessary and sufficient for the existence of such covers.
• The existence of such covers implies a surjection from $\pi_1(\mathbb{P}^2 \setminus (C_1 \cup C_2))$ onto the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$, as stated in Corollary 2.
• The condition that $C_2^+ - C_2^-$ has even intersection number with itself (i.e., $\nu \neq 0$) ensures that the existence of a single such cover for $n^2 \nmid \nu$ is sufficient for the full family to exist.
• The fundamental group of the complement of $C_1 \cup C_2$ surjects onto $\mathbb{F}_{1;(2,2,n_3,\dots,n_r)}$, and hence onto $\mathbb{Z}_2 * \mathbb{Z}_2$, due to the pencil structure.
• In special cases, such as the tricuspidal quartic or rational arrangements with multiple points, the fundamental group surjects onto $\mathbb{Z}_2 * \mathbb{Z}_2$ or $\mathbb{Z}_n * \mathbb{Z}_n$, confirming the general result.