A double covering of curves on a Hirzebruch surface of degree one and Weierstrass semigroups (Q666701)

Let $C$ be a smooth projective curve over $\mathbb{C}$, and $P$ a point on $C$. The Weierstrass semigroup $H(P)$ at $P$ is the set of non-negative integers $n$ such that there exists a meromorphic function $f$ on $C$ with $nP$ the pole of $f$ at $P$. The author investigates in this paper when a semigroup $H$ is the Weierstrass semigroup $H(\tilde {P})$ at a ramification point $\tilde {P}$ of a double covering $\pi : \tilde{C} \to C$ of a smooth curve $C$. The main result is Theorem 1.2, which has a very technical statement, allowing to relate $H(\tilde {P})$ and $H(P)$. A couple of standard examples show how to deal with the two cases arising in Theorem 1.2.