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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.

14H55 Riemann surfaces; Weierstrass points; gap sequences
14H45 Special algebraic curves and curves of low genus
14H50 Plane and space curves
Full Text: DOI
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