Faithfulness of Actions on Riemann-Roch Spaces
Faithfulness of Actions on Riemann-Roch Spaces
复制标题
黎曼-罗赫空间上作用的忠实性
DOI：
10.4153/cjm-2014-015-2
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Joseph Tait
中科院分区：
文献类型：
--
作者：
B. Köck;Joseph Tait
Abstract Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$ , we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$ -invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$ . This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ . Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.
Abstract Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$ , we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$ -invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$ . This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ . Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.