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On the topology of fiber-type curves: a Zariski pair of affine nodal curves

José Ignacio Cogolludo-Agustín, Eva Elduque

TL;DR

The paper addresses when the complement of a curve in a smooth projective surface has a fundamental group that is a free product of cyclic groups, with a focus on fiber-type curves in $\mathbb{P}^2$. It develops a framework based on admissible maps, orbifold fundamental groups, and threshold sets of atypical values to characterize when $\pi_1(\mathbb{P}^2\setminus D)$ is a free product, and it constructs infinite families of Zariski pairs (both affine and projective) that realize different embeddings while sharing tubular neighborhoods. A central finding is that the position of nodes can alter the topology of embeddings, and twisted Alexander polynomials with respect to finite $\mathrm{SU}(2)$ representations can distinguish such pairs even when abelian invariants coincide. The results yield a complete classification of threshold behavior in the studied example and provide new tools for distinguishing Zariski pairs via non-abelian invariants, with implications for the topology of plane curve complements.

Abstract

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb P^2$, then such curves must be of fiber type, i.e. a finite union of fibers of an admissible map onto a complex curve. In this setting, we exhibit an infinite family of Zariski pairs of fiber-type curves, that is, pairs of plane projective fiber-type curves whose tubular neighborhoods are homeomorphic, but whose embeddings in $\mathbb P^2$ are not. This includes a Zariski pair of curves in $\mathbb C^2$ with only nodes as singularities (and the same singularities at infinity) whose complements have non-isomorphic fundamental groups, one of them being free. Our examples show that the position of nodes also affects the topology of the embedding of projective curves. Twisted Alexander polynomials with respect to finite $SU(2)$ representations show to be useful for this purpose, since all their abelian invariants are the same for both fundamental groups.

On the topology of fiber-type curves: a Zariski pair of affine nodal curves

TL;DR

The paper addresses when the complement of a curve in a smooth projective surface has a fundamental group that is a free product of cyclic groups, with a focus on fiber-type curves in . It develops a framework based on admissible maps, orbifold fundamental groups, and threshold sets of atypical values to characterize when is a free product, and it constructs infinite families of Zariski pairs (both affine and projective) that realize different embeddings while sharing tubular neighborhoods. A central finding is that the position of nodes can alter the topology of embeddings, and twisted Alexander polynomials with respect to finite representations can distinguish such pairs even when abelian invariants coincide. The results yield a complete classification of threshold behavior in the studied example and provide new tools for distinguishing Zariski pairs via non-abelian invariants, with implications for the topology of plane curve complements.

Abstract

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is , then such curves must be of fiber type, i.e. a finite union of fibers of an admissible map onto a complex curve. In this setting, we exhibit an infinite family of Zariski pairs of fiber-type curves, that is, pairs of plane projective fiber-type curves whose tubular neighborhoods are homeomorphic, but whose embeddings in are not. This includes a Zariski pair of curves in with only nodes as singularities (and the same singularities at infinity) whose complements have non-isomorphic fundamental groups, one of them being free. Our examples show that the position of nodes also affects the topology of the embedding of projective curves. Twisted Alexander polynomials with respect to finite representations show to be useful for this purpose, since all their abelian invariants are the same for both fundamental groups.
Paper Structure (11 sections, 18 theorems, 33 equations, 1 figure)

This paper contains 11 sections, 18 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible maps
  4. Orbifold fundamental groups and morphisms
  5. Fiber-type curves
  6. Threshold of atypical values
  7. Zariski pairs of fiber-type curves
  8. A Zariski pair of affine nodal curves
  9. A projective Zariski pair
  10. Threshold sets of atypical values for $F$
  11. Twisted Alexander polynomials for $\mathop{\mathrm{SU}}\nolimits(2)$-representations

Key Result

Theorem 1.1

The affine nodal curves $D_1$ and $D_2$ form an affine Zariski pair. Moreover, $\pi_1(\mathbb{C}^2\setminus D_1)\not\cong\pi_1(\mathbb{C}^2\setminus D_2)=\mathbb{F}_2$ the free group of rank 2.

Figures (1)

  • Figure 1: Real curve $H_{B_1\setminus\{\infty\}}\cup \ell_1\cup\ell_2$

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1
  • proof
  • Example 2.1
  • ...and 42 more