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Equivalence of finite group actions on Riemann surfaces and algebraic curves
Contemporary Mathematics, accepted (2021)
  • Sean A Broughton
We consider conformal actions of the fi nite group G on a closed
Riemann surface S, as well as algebraic actions of G on smooth, complete,
algebraic curves over an arbitrary, algebraically closed field. There are several
notions of equivalence of actions, the most studied of which is topological
equivalence, because of its close relationship to the branch locus of moduli
space. A second important equivalence relation is that induced by representations
of G on spaces of holomorphic q-diff erentials. The notion of topological
equivalence does not work well in positive characteristic. We shall discuss an
alternative to topological equivalence, which we dub equisymmetry, that may
be applied in all characteristics. The relation is induced by families of curves
with G-action, and it works well with rotation constants and q-diff erentials,
which are also defi ned in positive characteristic. After giving an overview of the
various equivalence relations (conformal/algebraic, topological, q-di fferentials,
rotation constants, equisymmetry) we focus on the interconnections among
rotation constants, q-di fferentials, and equisymmetry.
  • Compact Riemann surface,
  • group action,
  • automorphism group,
  • differential
Publication Date
April 27, 2021
Citation Information
Sean A Broughton. "Equivalence of finite group actions on Riemann surfaces and algebraic curves" Contemporary Mathematics, accepted (2021)
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