Dimitri Markushevich ; Anne Moreau - Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions

epiga:11511 - Épijournal de Géométrie Algébrique, July 11, 2024, Volume 8 - https://doi.org/10.46298/epiga.2024.11511
Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functionsArticle

Authors: Dimitri Markushevich ; Anne Moreau

    Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.


    Volume: Volume 8
    Published on: July 11, 2024
    Accepted on: February 7, 2024
    Submitted on: June 28, 2023
    Keywords: Mathematics - Algebraic Geometry,14B05, 11F22, 20D06, 14H45, 20H15

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