Galois Lines for Space Curves

Abstract

Let C be a curve, and l, l0 be lines in the projective three space P 3 . Consider a projection π l : P 3 · · · → l0 with center l, where l ∩ l0 = ∅. Restricting π l to C, we get a morphism π l |C : C → l0 and an extension of fields (π l |C ) * : k(l0) → k(C). We study the algebraic structure of the extension and the geometric structure of C. In particular, we study the structure of the Galois group and the number of Galois lines for some special cases.

Galois subspaces for the rational normal curve

“…The purpose of this article is to classify all linear subspaces of P n that induce a Galois morphism for an embedding of P 1 into P n , thereby answering a question asked by Yoshihara in his list of open problems [Yos18]. This question was already addressed by Yoshihara himself for n = 3 in [Yos06,Prop. 4.1], and our article can be seen as a generalization of his results to arbitrary dimension.…”

Galois points for a plane curve in arbitrary characteristic

“…Duyaguit and Yoshihara [5,43] studied space curves in P 3 with a Galois line. The results have some applications to the study of plane curves.…”

Galois lines for the Artin-Schreier-Mumford curve

“…In 1996, Hisao Yoshihara introduced the notion of a Galois point: for a plane curve C ⊂ P 2 over an algebraically closed field k, a point P ∈ P 2 is called a Galois point if the function field extension k(C)/π * P k(P 1 ) induced by the projection π P from P is Galois ( [6,11]). Analogously, Yoshihara introduced the notion of a Galois line: a line ℓ ⊂ P 3 is said to be Galois for a space curve X ⊂ P 3 , if the extension k(X)/π * ℓ k(P 1 ) induced by the projection π ℓ : X P 1 from ℓ is Galois ( [1,12]). The following problem is raised in [13].…”