Report
Action of the automorphism group on the Jacobian of Klein's quartic curve
| العنوان: | Action of the automorphism group on the Jacobian of Klein's quartic curve |
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| المؤلفون: | Markushevich, Dimitri, Moreau, Anne |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, 14B05, 20D06, 14H45, 20H15, 11F22 |
| الوصف: | Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According to a conjecture of Bernstein--Schwarzman, a quotient of $\mathbb C^n$ by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture is known in dimension two and for complexifications of the real crystallographic groups generated by reflections. The case of $\tilde G$ is the first, and in a sense the smallest of the unknown cases. We compute the orbits and the stabilizers of the action of $G$ on $J$ and deduce that $J/G=\mathbf C^3/\tilde G$ is a strongly simply connected variety with the same singularities as the weighted projective space $\mathbb P(1,2,4,7)$. Comment: Minor changes; typos and small imprecisions corrected, 4 references added |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2107.03745 |
| رقم الانضمام: | edsarx.2107.03745 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |