Classifying compact Riemann surfaces by number of symmetries

Author:

Sebastián Reyes-Carocca, Pietro Speziali

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

date:

2023-10-10 16:00:00

Abstract

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is $3g$ and $3g+3,$ where $g$ is the genus. We prove that if $g$ is odd (respectively $g$ even and $g \not \equiv 2 \mbox{ mod } 3$) then there exists a unique Riemann surface of genus $g$ with a group of automorphisms of order $3g$ (respectively $3g+3$). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.

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