Finite quotients of powers of an elliptic curve

2019/05/16 by Chirvasitu, Alex, Kanda, Ryo, Smith, S. Paul
#14A22 #14E20 #14H52 #14L30 (Primary) #16S38 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1905.06710

Abstract

Let E be an elliptic curve. When the symmetric group Σ_g+1 of order (g+1)! acts on E^g+1 in the natural way, the subgroup E₀^g+1, consisting of those (g+1)-tuples whose coordinates sum to zero, is stable under the action of Σ_g+1. It is isomorphic to E^g. This paper concerns the structure of the quotient variety E^g/Σ when Σ is a subgroup of Σ_g+1 generated by simple transpositions. In an earlier paper we observed that E^g/Σ is a bundle over a suitable power, E^N, with fibers that are products of projective spaces. This paper shows that E^g/Σ has an étale cover by a product of copies of E and projective spaces with an abelian Galois group.

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