On the number of Galois points for a plane curve in characteristic zero

TL;DR

This paper establishes upper bounds on the number of Galois points for plane curves with positive genus in characteristic zero, confirming Yoshihara's conjecture in specific cases.

Contribution

It provides new upper bounds for Galois points on plane curves, especially when the curve is not immersed or has degree not divisible by two or three.

Findings

At most two Galois points if the curve is not immersed.

Outer Galois points are at most three if degree is not divisible by 2 or 3.

Yoshihara's conjecture is confirmed in these cases.

Abstract

For a plane curve, a point on the projective plane is said to be Galois if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. We present upper bounds for the number of Galois points, if the genus is greater than zero. If the curve is not an immersed curve, then we have at most two Galois points. If the degree is not divisible by two nor three, then the number of outer Galois points is at most three. As a consequence, a conjecture of Yoshihara is true in these cases.