On extensions of double coverings of plane curves and Weierstrass semigroups
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Let $C$ be a smooth plane curve of degree $d\geqq 4$ and $P$ be a point on $C$. Let $\pi:\tilde{C}\rightarrow C$ be a double covering of $C$ with a ramification point $\tP$ over $P$. We consider to determine a possible necessary condition for the Weierstrass semigroup of $\tP$ such that $\tC$ is contained in a K3 surface. Let $\nu$ be the multiplicity at $P$ of the intersection divisor between $C$ and the tangent line at $P$ on $C$. If $\nu=d$, then this problem was solved in \cite{ko-wa}. In this paper we also solve the problem in the case $\nu=d-1$.
Abstract
Let $C$ be a smooth plane curve of degree $d\geqq 4$ and $P$ be a point on $C$. Let $\pi:\tilde{C}\rightarrow C$ be a double covering of $C$ with a ramification point $\tP$ over $P$. We consider to determine a possible necessary condition for the Weierstrass semigroup of $\tP$ such that $\tC$ is contained in a K3 surface. Let $\nu$ be the multiplicity at $P$ of the intersection divisor between $C$ and the tangent line at $P$ on $C$. If $\nu=d$, then this problem was solved in \cite{ko-wa}. In this paper we also solve the problem in the case $\nu=d-1$.