RELATING GALOIS POINTS TO WEAK GALOIS WEIERSTRASS POINTS THROUGH DOUBLE COVERINGS OF CURVES
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The point $P \in \mathbb{P}^2$ is referred to as a Galois point for a nonsingular plane algebraic curve $C$ if the projection $\pi_P: C \rightarrow \mathbb{P}^1$ from $P$ is a Galois covering. In contrast, the point $P' \in C'$ is referred to as a weak Galois Weierstrass point of a nonsingular algebraic curve $C'$ if $P'$ is a Weierstrass point of $C'$ and a total ramification point of some Galois covering $f:C' \rightarrow \mathbb{P}^1$. In this paper, we discuss the following phenomena. For a nonsingular plane curve $C$ with a Galois point $P$ and a double covering $\varphi: C \rightarrow C'$, if there exists a common ramification point of $\pi_P$ and $\varphi$, then there exists a weak Galois Weierstrass point $P' \in C'$ with its Weierstrass semigroup such that $H(P')=\langle r,2r - 1 \rangle$ or $\langle r,2r + 1 \rangle$, which is a semigroup generated by two positive integers $r$ and $2r+1$ or $2r-1$, such that $P'$ is a branch point of $\varphi$. Conversely, for a weak Galois Weierstrass point $P' \in C'$ with $H(P')=\langle r,2r - 1 \rangle$ or $\langle r,2r + 1 \rangle$, there exists a nonsingular plane curve $C$ with a Galois point $P$ and a double covering $\varphi: C \rightarrow C'$ such that $P'$ is a branch point of $\varphi$.
Abstract
The point $P \in \mathbb{P}^2$ is referred to as a Galois point for a nonsingular plane algebraic curve $C$ if the projection $\pi_P: C \rightarrow \mathbb{P}^1$ from $P$ is a Galois covering. In contrast, the point $P' \in C'$ is referred to as a weak Galois Weierstrass point of a nonsingular algebraic curve $C'$ if $P'$ is a Weierstrass point of $C'$ and a total ramification point of some Galois covering $f:C' \rightarrow \mathbb{P}^1$. In this paper, we discuss the following phenomena. For a nonsingular plane curve $C$ with a Galois point $P$ and a double covering $\varphi: C \rightarrow C'$, if there exists a common ramification point of $\pi_P$ and $\varphi$, then there exists a weak Galois Weierstrass point $P' \in C'$ with its Weierstrass semigroup such that $H(P')=\langle r,2r - 1 \rangle$ or $\langle r,2r + 1 \rangle$, which is a semigroup generated by two positive integers $r$ and $2r+1$ or $2r-1$, such that $P'$ is a branch point of $\varphi$. Conversely, for a weak Galois Weierstrass point $P' \in C'$ with $H(P')=\langle r,2r - 1 \rangle$ or $\langle r,2r + 1 \rangle$, there exists a nonsingular plane curve $C$ with a Galois point $P$ and a double covering $\varphi: C \rightarrow C'$ such that $P'$ is a branch point of $\varphi$.
참고문헌 (14)
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