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Quasi-projective varieties whose fundamental group is a free product of cyclic groups

José Ignacio Cogolludo-Agustín, Eva Elduque

TL;DR

This work characterizes smooth complex quasi-projective surfaces whose fundamental group is a finite free product of cyclic groups by linking such groups to geometric fibrations onto curves via admissible maps. The authors develop a robust framework of orbifold fundamental groups, fiber-type decompositions, and Albanese-type constructions, yielding addition–deletion lemmas that control how removing or adding fibers alters the group structure. They prove a main structure theorem showing that, under suitable conditions, the surface admits an admissible map to a curve whose orbifold fundamental group matches the surface’s, with the divisor split into fiber-type and trivial-meridian parts; in the projective plane case they provide a stronger description and explicit pencils realizing the isomorphism. These results unify and extend classical plane-curve phenomena (notably $C_{p,q}$-type curves and torus-type sextics), and supply new, braid-monodromy-free proofs and constructions of curves in projective surfaces with prescribed free-product fundamental groups, highlighting the pervasiveness of such groups in quasi-projective topology.

Abstract

In this work we study smooth complex quasi-projective surfaces whose fundamental group is a free product of cyclic groups. In particular, we prove the existence of an admissible map from the quasi-projective surface to a smooth complex quasi-projective curve. Associated with this result, we prove addition-deletion Lemmas for fibers of the admissible map which describe how these operations affect the fundamental group of the quasi-projective surface. Our methods also allow us to produce curves in smooth projective surfaces whose fundamental groups of their complements are free products of cyclic groups, generalizing classical results on $C_{p,q}$ curves and torus type projective sextics, and showing how general this phenomenon is.

Quasi-projective varieties whose fundamental group is a free product of cyclic groups

TL;DR

This work characterizes smooth complex quasi-projective surfaces whose fundamental group is a finite free product of cyclic groups by linking such groups to geometric fibrations onto curves via admissible maps. The authors develop a robust framework of orbifold fundamental groups, fiber-type decompositions, and Albanese-type constructions, yielding addition–deletion lemmas that control how removing or adding fibers alters the group structure. They prove a main structure theorem showing that, under suitable conditions, the surface admits an admissible map to a curve whose orbifold fundamental group matches the surface’s, with the divisor split into fiber-type and trivial-meridian parts; in the projective plane case they provide a stronger description and explicit pencils realizing the isomorphism. These results unify and extend classical plane-curve phenomena (notably -type curves and torus-type sextics), and supply new, braid-monodromy-free proofs and constructions of curves in projective surfaces with prescribed free-product fundamental groups, highlighting the pervasiveness of such groups in quasi-projective topology.

Abstract

In this work we study smooth complex quasi-projective surfaces whose fundamental group is a free product of cyclic groups. In particular, we prove the existence of an admissible map from the quasi-projective surface to a smooth complex quasi-projective curve. Associated with this result, we prove addition-deletion Lemmas for fibers of the admissible map which describe how these operations affect the fundamental group of the quasi-projective surface. Our methods also allow us to produce curves in smooth projective surfaces whose fundamental groups of their complements are free products of cyclic groups, generalizing classical results on curves and torus type projective sextics, and showing how general this phenomenon is.
Paper Structure (22 sections, 34 theorems, 50 equations, 1 figure, 1 table)

This paper contains 22 sections, 34 theorems, 50 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible maps on $X$
  4. Fundamental groups of quasi-projective varieties.
  5. Homology of the complement
  6. Orbifold fundamental groups and orbifold admissible maps
  7. Fundamental groups of complements of fiber-type curves
  8. Characteristic Varieties
  9. Iitaka's (Quasi)-Albanese varieties
  10. Main theorem
  11. Proof of Theorem \ref{['thm:main:intro']}
  12. Extensions of the main theorem to finite cyclic groups
  13. Main Theorem for curves in $\mathbb{P}^2$
  14. Addition-Deletion Lemmas
  15. Preparation Lemmas
  16. ...and 7 more sections

Key Result

Corollary 1.1

If a curve complement in $\mathbb{P}^2$ has fundamental group $\mathbb{Z}_p*\mathbb{Z}_q$, with $p,q\in\mathbb{Z}_{>1}$ coprime integers, then the curve is given by a polynomial equation of the form $f_p^q+f_q^p=0$, for some $f_p$ and $f_q$ homogeneous polynomials in $\mathbb{C}[x,y,z]$ of degrees $

Figures (1)

  • Figure 1: Resolution of indeterminacy

Theorems & Definitions (94)

  • Corollary 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Generic Addition-Deletion Lemma
  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 84 more