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# Faithfulness of Actions on Riemann-Roch Spaces

Published:2014-06-19

• Bernhard Köck,
Mathematical Sciences , University of Southampton , Southampton SO17 1TJ , United Kingdom
• Joseph Tait,
Mathematical Sciences , University of Southampton , Southampton SO17 1TJ , United Kingdom
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## Abstract

Given a faithful action of a finite group $G$ on an algebraic curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least~$2g_X-2$. This leads to a concise answer to the question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes m})$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we furthermore provide an explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of~$G$ on the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
 Keywords: faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology
 MSC Classifications: 14H30 - Coverings, fundamental group [See also 14E20, 14F35] 30F30 - Differentials on Riemann surfaces 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14D15 - Formal methods; deformations [See also 13D10, 14B07, 32Gxx] 11R32 - Galois theory

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