Faithfulness of Actions on Riemann-Roch Spaces

Abstract

Abstract. Given a faithful action of a finite group G on an algebraic curve X of genus g X ≥ 2, 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 X − 2. This leads to a concise answer to the question when the action of G on the space H 0 (X, Ω ⊗m X ) of global holomorphic polydifferentials of order m is faithful. If X is hyperelliptic, we furthermore provide an explicit basis of H 0 (X, Ω ⊗m X ). 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.

Galois structure of the holomorphic differentials of curves

“…It follows from [4, §2] that D gl has a versal deformation ring in the sense of [40] and that the tangent space t D gl of D gl is isomorphic to the equivariant cohomology H 1 (G, T X ) of (X, G) where T X is the tangent sheaf on X. Using the arguments in [31, §3] and in [30, §7], we obtain, as in the beginning of the proof of [30,Thm. 7.1], that…”

“…2.3] from cyclic p-groups to p-hypo-elementary groups. In [30,Thm. 7.1], the k-dimension of t D gl has been determined for any finite group G satisfying the additional assumption that dim k M G = dim k M G for all finitely generated kG-modules M .…”

Automorphisms and the canonical ideal

“…which is known to be faithful, when X is not hyperelliptic and p = 2, see [14]. The representation ρ in turn gives rise to a series of representations…”

“…Definition 13. Choose any term order ≺ t for the variables {ω N,µ : (N, µ) ∈ A} and define the term order ≺ on the monomials of S as follows: (14) ω N1,µ1 ω N2,µ2…”

Automorphisms of algebraic curves

“…Consider a complete non-singular non-hyperelliptic curve of genus g ≥ 3 over an algebraically closed field K. The automorphism group of the ambient space P g−1 is known to be PGL g (k), [22, example 7.1.1 p. 151]. On the other hand every automorphism of X is known to act on H 0 (X, Ω X ) giving rise to a representation ρ : G → GL(H 0 (X, Ω X )), which is known to be faithful, when X is not hyperelliptic and p = 2, see [29]. The representation ρ in turn gives rise to a series of representations…”