Geometric realizability of epimorphisms to curve orbifold groups
José I. Cogolludo-Agustín, Eva Elduque
TL;DR
This work generalizes the Geometric Realizability Problem to orbifold curve groups by linking algebraic quotients of $\pi_1(U)$ with admissible maps to orbifold curves. The authors prove a sharp dichotomy: when the curve orbifold group has negative orbifold Euler characteristic $\chi_{g,(r,\bar{m})}<0$, every epimorphism $\pi_1(U)\to \mathbb{G}_{g,(r,\bar{m})}$ with finitely generated kernel is geometrically realized by a unique admissible map $F:U\to C$, with $F_*$ matching the quotient up to isomorphism; for $\chi_{g,(r,\bar{m})}\ge 0$ no such general realization holds, and explicit counterexamples are given. The paper develops the necessary preliminaries on Zariski open sets, characteristic varieties, orbifolds, the $ninf$ property, and the exact sequence associated to admissible maps, and then leverages these to address Serre’s question in $\mathbb{P}^2$, classifying realizability for many curve orbifold groups. The results deepen understanding of when quasi-projective fundamental groups arise as orbifold curve groups, and provide constructive methods (via orbifold pencils) to realize them in geometric settings with potential applications to plane curve complements.
Abstract
Given a connected dense Zariski open set of a compact Kähler manifold $U$, we address the general problem of the existence of surjective holomorphic maps ${F:U\to C}$ to smooth complex quasi-projective curves from properties of $π_1(U)$. It is known that, if such $F$ exists, then there exists a finitely generated normal subgroup $K\trianglelefteqπ_1(U)$ such that $π_1(U)/K$ is isomorphic to a curve orbifold group $G$ (i.e. the orbifold fundamental group of a smooth complex quasi-projective curve endowed with an orbifold structure). In this paper, we address the converse of that statement in the case where the orbifold Euler characteristic of $G$ is negative, finding a (unique) surjective holomorphic map $F:U\to C$ which realizes the quotient $π_1(U)\twoheadrightarrow π_1(U)/K\cong G$ at the level of (orbifold) fundamental groups. We also prove that our theorem is sharp, meaning that the result does not hold for any curve orbifold group with non-negative orbifold Euler characteristic. Furthermore, we apply our main theorem to address Serre's question of which orbifold fundamental groups of smooth quasi-projective curves can be realized as fundamental groups of complements of curves in $\mathbb{P}^2$.