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Criteria for the existence of a plane model with two Galois points for algebraic curves

Higashine Kazuki 山形大学

2021.03.31

概要

Let k be an algebraically closed field of characteristic p ≥ 0. We fix
k as the ground field of our discussion in this dissertation. In 1996, Hisao
Yoshihara introduced the notion of Galois points in algebraic geometry. Let
C be an irreducible (possibly singular) plane (algebraic) curve over k. We
consider the projection
πP : C 99K P1 ; Q 7→ P Q
with the center P ∈ P2 , where P Q represents the line passing through points
P , Q ∈ P2 if P ̸= Q. If the field extension k(C)/πP∗ k(P1 ) of function fields
induced by πP is Galois, then P is called a Galois point for C (see [9, 41, 47]).
Assume that P is a Galois point. If P is a smooth point of C (resp. a singular
point of C, a point contained in C, a point not contained in C), then P is
called a smooth Galois point (resp. a non-smooth Galois point, an inner
Galois point, an outer Galois point), after [39, 40, 45]. The associated Galois
group
GP = Gal(k(C)/πP∗ k(P1 ))
is called a Galois group at P .
In the theory of Galois points, plane curves with two or more Galois points
are important. In 2013, with the contribution of four researchers Yoshihara,
Kei Miura, Masaaki Homma, and Satoru Fukasawa, a complete classification
of smooth plane curves with two or more Galois points was obtained ([41,
47, 36, 6, 8, 7, 10, 15, 12]). A classification of plane curves with infinitely
many inner Galois points was obtained by Fukasawa and Takehiro Hasegawa
[21]. ...

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57

...

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