Singular plane curves with infinitely many Galois points

Abstract

For a plane curve C , we call a point P ∈ P 2 a Galois point with respect to C if the point projection from P induces a Galois extension of function fields. We give an example of a singular plane curve having infinitely many inner and outer Galois points. We also classify plane curves whose general points are inner Galois points. Before our results, known examples in the theory of Galois points have only finitely many Galois points, except trivial cases.

An upper bound for the number of Galois points for a plane curve

“…Recently, the present author [3] showed that δ(C) = 0, 1 or d for any other smooth curve C. As a next step, it would be nice to give an upper bound for δ(C) for all irreducible plane curves C. Miura [16] gave a certain inequality related to δ(C) if p = 0 and d − 1 is prime. The present author and T. Hasegawa [6] settled the case δ(C) = ∞. We call this case (FH).…”

Galois points for a plane curve in arbitrary characteristic

“…In [10] it is also proven that (C) is a non-empty Zariski open set of C if and only if C is projectively equivalent to the curve defined by X Z q−1 − Y q . More recently, in [9] plane curves with infinitely many outer Galois points are also classified.…”

“…Recently, Fukasawa and Hasegawa [10] found an example having infinitely many inner and outer Galois points.…”

Algebraic curves admitting inner and outer Galois points

“…x − y q = 0, where p > 0 and q is a power of p. Let P = (0 : 0 : 1) and let Q ∈ {Z = 0} \ {(1 : 0 : 0), (0 : 1 : 0)}. According to [4], P is an inner Galois point and Q is an outer Galois point. Note that there exists an inclusion G P ֒→ P GL(3, k) and G P fixes Q.…”