New examples of Weierstrass semigroups associated with a double covering of a curve on a Hirzebruch surface of degree one

Abstract

Let $\varphi:\Sigma_1\longrightarrow \mathbb{P}^2$ be a blow up at a point on $\mathbb{P}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\geq 4$ by $\varphi$, and let $P$ be a point on $C$. Let $\pi:\tilde{C}\longrightarrow C$ be a double covering branched along the reduced divisor on $C$ obtained as the intersection of $C$ and a reduced divisor in $|-2K_{\Sigma_1}|$ containing $P$. In this paper, we investigate the Weierstrass semigroup $H(\tilde{P})$ at the ramification point $\tilde{P}$ of $\pi$ over $P$, in the case where the intersection multiplicity at $\varphi(P)$ of $\varphi(C)$ and the tangent line at $\varphi(P)$ of $\varphi(C)$ is $d-1$.