A birational embedding of an algebraic curve into a projective plane with two Galois points

A criterion for the existence of a plane model with two inner Galois points for algebraic curves

“…a singular point of C, a point contained in C, a point not contained in C), after [11,12,15]. Note that the definition of an inner Galois point in [5] is equivalent to the definition of a smooth Galois point in this article.…”

“…In 2016, Fukasawa [5] presented a criterion for the existence of a birational embedding of a smooth projective curve into a projective plane with two smooth Galois points and obtained new examples of plane curves with two smooth Galois points by using this criterion. On the other hand, there have been some known examples of plane curves with two or more non-smooth Galois points.…”

“…In this article, we extend Fukasawa's criterion [5,Theorem 1] to all cases with two (possibly non-smooth) Galois points. Let X be a (reduced, irreducible) smooth projective curve over k, and let kðX Þ be its function field.…”

A birational embedding with two Galois points for certain Artin–Schreier curves

“…Let C be a (reduced, irreducible) smooth projective curve over an algebraically closed field k of characteristic p ≥ 0 and let k(C) be its function field. Recently, a criterion for the existence of a birational embedding with two Galois points was presented by the first author ( [1]), and by this criterion, several new examples of plane curves with two Galois points were described. We recall this criterion.…”

“…A similar result holds for "outer" Galois points. In this case, we consider a 4-tuple (G 1 , G 2 , H, Q) with Q ∈ C such that G 1 and G 2 satisfy conditions (a') and (b'), and σ∈G 1 σ(Q) = τ ∈G 2 τ (Q) holds (see also [1,Remark 1]).…”