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 \(d\ge 4\) 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 \(d-1\).