On fundamental groups related to degeneratable surfaces: conjectures and examples

Friedman, Michael; Teicher, Mina

Friedman, Michael; Teicher, Mina

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, p. 565-603

Abstract

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

Published online : 2019-02-22

MR 3059838 | Zbl 1298.14015

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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