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

2024/02/16 by Chen, William Y., Salter, Nick · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2402.11081

Abstract

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\rightarrowAut⁺(F₂)→ 1 of Dyer-Formanek-Grossman.

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• The Yang-Baxter Equation and Characteristic Finite Simple Quotients of the Free Group of Rank 2

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