WEIERSTRASS POINTS WITH FIRST NON-GAP FOUR ON A DOUBLE COVERING OF A HYPERELLIPTIC CURVE

Abstract. Let H be a 4-semigroup, i.e., a numerical semigroup whose minimum positive element is four. We denote by 4r(H) + 2 the minimum element of H which is congruent to 2 modulo 4. If the genus g of H is larger than 3r(H) − 1, then there is a cyclic covering pi: C − → P1 of curves with degree 4 and its ramification point P such that the Weierstrass semigroup H(P) of P is H (Komeda [1]). In this paper it is showed that we can construct a double covering of a hyperelliptic curve and its ramification point P such that H(P) is equal to H even if g ≤ 3r(H) − 1