On \(\gamma\)-hyperelliptic numerical semigroups. (English) Zbl 0931.14017

Torres, Fernando

Semigroup Forum 55, No. 3, 364-379 (1997).

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

Keywords:

\(\gamma\)-hyperelliptic numerical semigroup; Weierstrass semigroups