Abstract view
Faithfulness of Actions on RiemannRoch Spaces


Published:20140619
Printed: Aug 2015
Bernhard Köck,
Mathematical Sciences , University of Southampton , Southampton SO17 1TJ , United Kingdom
Joseph Tait,
Mathematical Sciences , University of Southampton , Southampton SO17 1TJ , United Kingdom
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 RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. 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, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology
faithful action, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology

MSC Classifications: 
14H30, 30F30, 14L30, 14D15, 11R32 show english descriptions
Coverings, fundamental group [See also 14E20, 14F35] Differentials on Riemann surfaces Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] Formal methods; deformations [See also 13D10, 14B07, 32Gxx] Galois theory
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
