Toric varieties - degenerations and fundamental groups

TL;DR

This paper computes the fundamental groups of complements of four toric varieties' branch curves, revealing their structure as quotients of braid groups and concluding that their Galois covers are simply connected.

Contribution

It provides explicit calculations of fundamental groups for specific toric varieties and explores their relations to braid groups and degenerations, advancing understanding of their topological properties.

Findings

Fundamental groups are quotients of Artin braid groups.

Galois covers of the varieties are simply connected.

Explicit relations between toric degenerations and braid monodromies.

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.