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

Let $\phi :{\mathrm{\Sigma }}_1⟶{\mathbb{P}}^2$ be a blow up at a point on ${\mathbb{P}}^2$. Let $C$ be the proper transform of a smooth plane curve of degree $d\ge 4$ by $\phi$, and let $P$ be a point on $C$. Let $\pi :\tilde{C}⟶C$ be a double covering branched along the reduced divisor on $C$ obtained as the intersection of $C$ and a reduced divisor in $\mathrm{\mid }-2K_{{\mathrm{\Sigma }}_1}\mathrm{\mid }$ 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 $\phi (P)$ of $\phi (C)$ and the tangent line at $\phi (P)$ of $\phi (C)$ is $d-1$.