Faithfulness of actions on Riemann-Roch spaces
Faithfulness of actions on Riemann-Roch spaces
Given a faithful action of a finite group G on an algebraic curve X of genus g > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g-2. This leads to a concise answer to the question when the action of G on the space H^0(X, \Omega_X^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^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, Z/mZ) if X is a Riemann surface.
848-869
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Tait, Joseph
08f47697-9060-4958-b69a-facdafc478bc
22 June 2015
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Tait, Joseph
08f47697-9060-4958-b69a-facdafc478bc
Koeck, Bernhard and Tait, Joseph
(2015)
Faithfulness of actions on Riemann-Roch spaces.
Canadian Journal of Mathematics, 67 (4), .
(doi:10.4153/CJM-2014-015-2).
Abstract
Given a faithful action of a finite group G on an algebraic curve X of genus g > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g-2. This leads to a concise answer to the question when the action of G on the space H^0(X, \Omega_X^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^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, Z/mZ) if X is a Riemann surface.
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Accepted/In Press date: 11 April 2014
Published date: 22 June 2015
Organisations:
Pure Mathematics
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Local EPrints ID: 364271
URI: http://eprints.soton.ac.uk/id/eprint/364271
ISSN: 0008-414X
PURE UUID: 86d49642-3b6f-433c-8ffd-6b2d64c44466
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Date deposited: 15 Apr 2014 11:51
Last modified: 15 Mar 2024 03:10
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Author:
Joseph Tait
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