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 C with barycenter 0 is isomorphic to the space of triples (E,Q,ω), where E is an elliptic curve, Q∈ E^∘ 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 B₄: namely, the sequence 1→ F₂→ B₄→ B₃→ 1, where F₂ is a free group of rank 2, related to Ferrari's solution of the quartic, and the sequence 1→ Z → B₄⁺(F₂)→ 1 of Dyer-Formanek-Grossman.