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

Let φ :Σ _1⟶ℙ^2 be a blow up at a point on ℙ^2 . Let C be the proper transform of a smooth plane curve of degree d≥ 4 by φ , and let P be a point on C . Let π :C̃⟶ C be a double covering branched along the reduced divisor on C obtained as the intersection of C and a reduced divisor in |-2K_Σ _1| containing P . In this paper, we investigate the Weierstrass semigroup H(P̃) at the ramification point P̃ of π over P , in the case where the intersection multiplicity at φ (P) of φ (C) and the tangent line at φ (P) of φ (C) is d-1 .