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

Cogolludo-Agustín, José I. (Universidad de Zaragoza) ; Elduque, Eva

Resumen: 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 P2, 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 P2 are not. This includes a Zariski pair of curves in C2 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.

Idioma: Inglés

DOI: 10.1007/s13398-026-01853-1

Año: 2026

Publicado en: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas 120, 3 (2026), [19 pp.]
ISSN: 1578-7303

Financiación: info:eu-repo/grantAgreement/ES/DGA/E22-20R
Financiación: info:eu-repo/grantAgreement/ES/MCIU/PID2024-156181NB-C33
Financiación: info:eu-repo/grantAgreement/ES/MICINN/RYC2021-031526-I

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Geometría y Topología (Dpto. Matemáticas)