On extensions of a double covering of plane curves and Weierstrass semigroups of the double covering type

Abstract

Let $C$ be a smooth plane curve of degree $d, P$ be a point on $C$, and let $\pi :\tilde{C}\rightarrow C$ be a double covering of the curve $C$ with the branch point $P$. In this paper, we give a best possible sufficient condition for the double covering $\pi$ to extend to a double covering $\tilde{\pi }:X\rightarrow {\mathbb {P}}^{2}$ branched along a reduced divisor of degree six which intersects transversally the curve $C$ at smooth $6d$ points containing $P$.