Dihedral coverings of trigonal curves (Q2841845)

After giving basic notions and facts related to trigonal curves in Hirzebruch surfaces, the author discusses the braid monodromy and focuses on the Zariski-van Kampen theorem and its implications for the particular case of trigonal curves; he then introduces the concept of universal trigonal curves related to a prescribed monodromy group or a prescribed quotient of \({\pi}^{afn}\) and proves their existence. Then he considers the geometric properties and fundamental groups of the universal curves corresponding to uniform dihedral coverings, obtaining as a consequence some restrictions on the fundamental group of a plane curve \(D \) with a singularity of multiplicity \((\deg(D) - 3)\). Some applications of the principal results are: the relation to the Mordell-Weil group, Z-splitting sections of trigonal curves, extensions to generalized trigonal curves and plane curves with deep singularities.