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

We show that the configuration space of four unordered points in $\mathbb{C}$ with barycenter 0 is isomorphic to the space of triples $(E,Q,\omega )$, where $E$ is an elliptic curve, $Q\in E^{\circ }$ a nonzero point, and $\omega$ 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 $1\to F_2\to B_4\to B_3\to 1$, where $F_2$ is a free group of rank 2, related to Ferrari’s solution of the quartic, and the sequence $1\to \mathbb{Z}\to B_4\to {\mathrm{Aut}}^{+}(F_2)\to 1$ of Dyer-Formanek-Grossman.