Hurwitz' problem on Weierstrass points on algebraic curves

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We are interested in Hurwitz' Problem which is the following : Find a necessary and sufficient computable condition on a numerical semigroup to be attained by a pointed curve, i. e., to be Weierstrass. This problem is not solved yet. But we got the following results : (1) Not every numerical semigroup whose minimum integer is eight(resp. twelve) is Weierstrass. (2) For a numerical semigroup H we denote by d(H) the numerical semigroup consisting of h/2 for even integer h in H. If the genus of d(H) is either 3 or 4 and the genus of H is larger than or equal to three times the genus of d(H), then H is Weierstrass. (3) We showed the following : If the genus of d(H) is larger than or equal to 5, the statement like(2) does not hold.