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On $$\gamma$$-hyperelliptic numerical semigroups. (English) Zbl 0931.14017
Let $$H$$ be a numerical semigroup. The genus of $$H$$ is the number of elements of the complement of $$H$$ in the natural numbers. $$H$$ is called $$\gamma$$-hyperelliptic if $$H$$ has $$\gamma$$ even elements in $$[2,4\gamma]$$, and the $$(\gamma+1)$$-th element is $$4\gamma+2$$. For a point $$P$$ of projective, irreducible, non-singular algebraic curve $$X$$ over an algebraically closed field $$k$$, let $$H(P)$$ be the set of orders $$m_i$$ at $$P$$ of meromorphic functions on $$X$$ with only pole at $$P$$, called the Weierstrass semigroup of $$X$$ at $$P$$, and $$w(P)$$ be the weight of $$P$$. There are numerical semigroups which are not Weierstrass semigroups.
The author extends the following equivalent results on Weierstrass semigroups $$H(P)$$ at ramified points of a double covering $$\pi:X$$ (of genus $$g)\to\widehat X$$ (of genus $$\gamma)$$ to any numerical semigroup $$H$$ of genus large enough:
1. $$H(P)$$ is $$\gamma$$-hyperelliptic if $$\text{char} (k)\neq 2$$ and $$g\geq 4\gamma+1$$, and $$g\geq 6\gamma-3$$ otherwise.
2. if $$g\geq 5\gamma+1$$, then $$m_{2\gamma+1} =6\gamma+ 2$$.
3. $$m_{g/2- \gamma-1}=g-2$$.
4. $${g-2\gamma \choose 2}\leq w(P)< {g-2\gamma+2 \choose 2}$$
to:
2’. if $$g\geq 6\gamma +4$$, then $$m_{2\gamma+1}(H)=6\gamma+2$$
or
2”. In the case $$g=6\gamma +5$$ or $$g\geq 6\gamma+7$$, if $$g$$ is even then $$m_r(H)=g-2$$, and if $$g$$ is odd, then $$m_r(H)=g-1$$, where $$r=[(g+1)/2]- \gamma-1$$.
3”. $$m_r(H) \leq g-1 <m_{r+1} (H)$$.
The author also improves results on weights of (Weierstrass) semigroups.
Reviewer: R.Horiuchi (Kyoto)

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences
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