ON EXTENSIONS OF DOUBLE COVERINGS OF PLANE CURVES AND WEIERSTRASS SEMIGROUPS

ON EXTENSIONS OF DOUBLE COVERINGS OF PLANE CURVES AND WEIERSTRASS SEMIGROUPS

Let C be a smooth plane curve of degree d ≧ 4 and P be a point on C. Let ${\pi}:{\tilde{C}}{\rightarrow}C$ be a double covering of C with a ramification point ${\tilde{P}}$ over P. We consider to determine a possible necessary condition for the Weierstrass semigroup of ${\tilde{P}}$ such that ${\tilde{C}}$ is contained in a K3 surface. Let ν be the multiplicity at P of the intersection divisor between C and the tangent line at P on C. If 𝜈 = d, then this problem was solved in [7]. In this paper we also solve the problem in the case 𝜈 = d - 1.

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