Toric varieties - degenerations and fundamental groups

Abstract

In this paper we calculate fundamental groups (and some of their quotients) of complements of four toric varieties branch curves. For these calculations, we study properties and degenerations of these toric varieties and the braid monodromies of the branch curves in $\mathbb{CP}^2$. The fundamental groups related to the first three toric varieties turn to be quotients of the Artin braid groups $\mathcal{B}_5$, $\mathcal{B}_6$, and $\mathcal{B}_4$, while the fourth one is a certain quotient of the group $\tilde{\mathcal{B}}_6 = \mathcal{B}_6/<[X,Y]>$, where $X, Y$ are transversal. The quotients of all four groups by the normal subgroups generated by the squares of the standard generators are respectively $S_5, S_6, S_4$ and $S_6$. We therefore conclude that the fundamental groups of the Galois covers of the four given toric varieties are all trivial.