Algebraic curves admitting inner and outer Galois points

TL;DR

This paper provides a criterion for when algebraic curves can be embedded into a projective plane with both inner and outer Galois points and classifies certain plane curves with specific Galois point group structures.

Contribution

It introduces a new criterion for birational embeddings with Galois points and classifies plane curves with particular Galois group configurations under certain characteristic assumptions.

Findings

Established a criterion for the existence of such embeddings.

Classified plane curves with Galois points where the groups form specific semi-direct products.

Focused on cases where the characteristic is zero or does not divide d-1.

Abstract

There are two purposes in this article. One is to present a criterion for the existence of a birational embedding into a projective plane with inner and outer Galois points for algebraic curves. Another is to classify plane curves of degree $d$ admitting an inner Galois point $P$ and an outer Galois point $Q$ with $G_PG_Q=G_P \rtimes G_Q$ or $G_P \ltimes G_Q$, under the assumption that the characteristic $p$ is zero or $p$ does not divide $d-1$.