Faithfulness of actions on RiemannRoch spaces
Abstract
Given a faithful action of a finite group G on an algebraic curve X of genus g_X > 1, we give explicit criteria for the induced action of G on the RiemannRoch space H^0(X,O_X(D)) to be faithful, where D is a Ginvariant 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^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.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.3135
 Bibcode:
 2014arXiv1404.3135K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14H30 (Primary) 30F30;
 14L30;
 14D15;
 11R32 (Secondary)
 EPrint:
 27 pages, 1 figure, to appear in Canadian Journal of Mathematics