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On the weight of numerical semigroups. (English) Zbl 1194.14048
Let $$S$$ be an irreducible, non-singular, algebraic projective curve of genus $$g \geq 2$$ defined over some algebraically closed field of characteristic zero. To each point $$p \in S$$ there is associated a semigroup $$H_{S}(p)$$, whose elements are the orders as pole at $$p$$ of regular functions on $$S-\{p\}$$. The integers of set $$G_{S}(p):={\mathbb N}_{0} \setminus H_{S}(p)=\{l_{1}< \cdots < l_{g}\}$$ are called the gaps of $$p$$. The Weierstrass gap theorem asserts that $$l_{g} \leq 2g-1$$. The weight of $$p$$ is defined by $$w(p)=\sum_{j=1}^{g}(l_{j}-1)$$.
Let $$\pi:S \to R$$ be a degree two morphism over another curve $$R$$ of genus $$\gamma$$. Assume that $$p \in S$$ is so that $$\pi(p)$$ is a branch value of $$\pi$$. If $$2H_{R}(\pi):=\{2h: h \in H_{R}(\pi(p))\}$$, then $$H_{S}(p)=2H_{R}(\pi(p)) \cup \{u_{1},\dots ,u_{\gamma}\} \cup \{2g+j:j \in {\mathbb N}_{0}\}$$, where $$u_{1},\dots ,u_{\gamma}$$ are the odd non-gaps in the set $$\{3,\dots ,2g-1\}$$. If $$F(\gamma)=\gamma^{2}+4\gamma+3$$ for $$\gamma \geq 7$$, $$F(0)=2$$, $$F(1)=11$$, $$F(2)=23$$, $$F(3)=34$$, $$F(4)=44$$, $$F(5)=56$$ and $$F(6)=65$$, then it is well known that $\binom{g-2\gamma}{2} \leq w(p) \leq \binom{g-2\gamma}{2}+2\gamma^{2}.$ The authors study the following problem: Describe the true values of $$w(p)$$ that may appear.
In order to study the above problem, the authors consider the collection of abstract semigroups of genus $$g \geq 2$$, that is, sets of the form $$H=\{0=m_{0}<m_{1}<m_{2}<\cdots\} \subset {\mathbb N}_{0}$$ so that $$G:={\mathbb N}_{0} \setminus H=\{l_{1}<\cdots<l_{g}\}$$. The weight of $$H$$ is defined by $$w(H)=\sum_{j=1}^{g}(l_{j}-1)$$. The semigroup $$H$$ is called $$\gamma$$-hyperelliptc if in the set $$G:={\mathbb N}_{0} \setminus H$$ there are exactly $$\gamma$$ even integers. Also, $$H$$ is said to satisfy the Weierstrass property if there exists some curve $$S$$ of genus $$g$$ and some point $$p \in S$$ so that $$H=H(p)$$.
Then the authors describe (with explicit examples) those abstract semigroups $$H$$ of genus $$g$$ satisfying $\binom{g-2\gamma}{2} \leq w(H) \leq \binom{g-2\gamma}{2}+2\gamma^{2}$ which also satisfy the Weierstrass property.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H45 Special algebraic curves and curves of low genus 14H50 Plane and space curves
##### Keywords:
algebraic curves; morphisms; semigroups
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##### References:
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