Hurwitz' problem on Weierstrass points on algebraic curves

我们感兴趣的是Hurwitz问题，它是：求一个数值半群的可计算性的充要条件是一条尖曲线，即：例如，成为维尔斯特拉斯这个问题还没有解决。但我们得到了如下结果：（1）不是每个最小整数为8的数值半群（分别为（12）Weierstrass。(2)对于一个数值半群H，我们用d（H）表示H中由h/2组成的数值半群，其中h为偶数。如果d（H）的亏格为3或4，且H的亏格大于或等于d（H）亏格的3倍，则H是Weierstrass。(3)我们证明了：如果d（H）的亏格大于或等于5，则像（2）这样的陈述不成立。

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.