On fundamental groups related to degeneratable surfaces: conjectures and examples

Friedman, Michael; Teicher, Mina

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 3, pp. 565-603.

We argue that for a smooth surface $S$, considered as a ramified cover over ${\mathrm{ℂ\mathbb{P}}}^2$, branched over a nodal-cuspidal curve $B\subset {\mathrm{ℂ\mathbb{P}}}^2$, one could use the structure of the fundamental group of the complement of the branch curve ${\pi }_1({\mathrm{ℂ\mathbb{P}}}^2-B)$ 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 ${\pi }_1({\mathrm{ℂ\mathbb{P}}}^2-B)$ 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 ${\mathrm{ℂ\mathbb{P}}}^1\times C_g$, where $C_g$ is a curve of genus $g$, ${\pi }_1({\mathrm{ℂ\mathbb{P}}}^2-B)$ is a quotient of an Artin group associated to the degeneration.

Classification : 14D06, 14Q10, 14H20, 14H30, 20F36

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