Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands

Abstract

We show that the configuration space of four unordered points in $\mathbb{C}$ with barycenter 0 is isomorphic to the space of triples $(E,Q,ω)$, where $E$ is an elliptic curve, $Q\in E^\circ$ a nonzero point, and $ω$ a nonzero holomorphic differential on $E$. At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group $B_4$: namely, the sequence $1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1$, where $F_2$ is a free group of rank 2, related to Ferrari's solution of the quartic, and the sequence $1\rightarrow \mathbb{Z} \rightarrow B_4\rightarrow\operatorname{Aut}^+(F_2)\rightarrow 1$ of Dyer-Formanek-Grossman.

Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Remark 2.5