On \(\gamma\)-hyperelliptic numerical semigroups
type: Item
instance of: scholarly article
title: On \(\gamma\)-hyperelliptic numerical semigroups
zbMATH Open document ID: 0931.14017
DOI: 10.1007/PL00005936
author: Fernando Torres
published in: Semigroup Forum
publication date: 2000-02-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 1106540
zbMATH Keywords: \(\gamma\)-hyperelliptic numerical semigroup
zbMATH Keywords: Weierstrass semigroups
reviewed by: Ryutaro Horiuchi
MaRDI profile type: Publication
full work available at URL: pl00005936
OpenAlex ID: W2319291441
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description: scientific article; zbMATH DE number 1106540