On fundamental groups related to degeneratable surfaces: conjectures and examples

TL;DR

This paper investigates the fundamental groups of degeneratable surfaces, confirming a conjecture for embedded-degeneratable surfaces and proposing new conjectures for non-embedded-degeneratable cases, with supporting theorems and specific examples.

Contribution

It proves Teicher's conjecture for embedded-degeneratable surfaces and introduces new conjectures on the fundamental groups of non-embedded-degeneratable surfaces.

Findings

Confirmed Teicher's conjecture for embedded-degeneratable surfaces.

Proposed two new conjectures for non-embedded-degeneratable surfaces.

Showed that for certain surfaces, the fundamental group is a quotient of an Artin group.

Abstract

We argue that for a smooth surface S, considered as a ramified cover over the projective plane branched over a nodal-cuspidal curve B one could use the structure of the fundamental group of the complement of the branch curve to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces - a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of the mentioned fundamental group for these surfaces and present two new conjectures on the structure of this group, regarding non-embedded-degeneratable surfaces. We prove two theorems supporting our conjectures, and show that for an empbedding of a product of a projective line with a curve of genus g, the fundamental group of the complement of the branch curve is a quotient of an Artin group associated to the degeneration.