On fundamental groups related to degeneratable surfaces: conjectures and examples
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, p. 565-603

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

Published online : 2019-02-22
Classification:  14D06,  14Q10,  14H20,  14H30,  20F36
@article{ASNSP_2012_5_11_3_565_0,
author = {Friedman, Michael and Teicher, Mina},
title = {On fundamental groups related to degeneratable surfaces: conjectures and examples},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {3},
year = {2012},
pages = {565-603},
zbl = {1298.14015},
mrnumber = {3059838},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_3_565_0}
}

Friedman, Michael; Teicher, Mina. On fundamental groups related to degeneratable surfaces: conjectures and examples. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 565-603. http://www.numdam.org/item/ASNSP_2012_5_11_3_565_0/

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