Weierstrass points and double coverings of curves. With application: Symmetric numerical semigroups which cannot be realized as Weierstrass semigroups (Q1328176)

Let \(X\) be a nonsingular projective curve of genus \(\gamma\geq 1\); such curves are called \(\gamma\)-hyperelliptic curves. For a point \(P\) of \(X\), let \(H(P)\) be the Weierstrass semigroup of \(X\) at \(P\), and \(w(P)\) be the weight of \(P\). A numerical sub-semigroup \(H\) of \((\mathbb{N},+)\) is said to be \(\gamma\)-hyperelliptic if the first \(\gamma\) positive terms \(M_1, \ldots, M_\gamma\) are even, \(M_\gamma= 4\gamma\), and \(4\gamma + 2 \in H\). The author proves with \(g \geq 6 \gamma + 4\) the equivalence of the three properties: \(\gamma\)-hyperellipticity of \(X\), the existence of a point \(P \in X\) such that \(H(P)\) is \(\gamma\)-hyperelliptic, and the existence of a complete, base-point-free linear system on \(X\) of projective dimension \(2 \gamma + 1\) and degree \(6 \gamma + 2\). He also proves the equivalence of \(\gamma\)-hyperellipticity of \(X\) and the existence of a point \(P \in X\) such that \({g - 2 \gamma \choose 2} \leq w(P) \leq {g - 2 \gamma \choose 2} + 2 \gamma^2\), where \(g\geq 30\) if \(\gamma=1\), and \(g\geq {{12\gamma-6} \choose 2}+1\) if \(\gamma\geq 2\). These improves results by \textit{T. Kato}, \textit{J. Komeda} and \textit{A. Garcia}. As a by-product, the author shows how to construct \(\gamma\)-hyperelliptic symmetric (i.e., \(2g - 1\) is a gap) numerical semigroups which are not Weierstrass semigroups. The first example of a (non-symmetric) numerical semigroup which is not a Weierstrass semigroup was given by R. O. Buchweitz. It was generalized by \textit{J. Komeda} [see ``Numerical semigroups and non-gaps of Weierstrass points'', Res. Rep. Ikutoku Tech. Univ. B-9 (1985), ``On non-Weierstrass gap sequences'', Res. Rep. Kanagawa Inst. Technology, B-13 (1989), ``Non-Weierstrass numerical semigroups'' (Preprint)] and by \textit{H. Ishida}, \textit{T. Kato} and the reviewer [see ``A note on Buchweitz gap sequences'', Acta Human. Sci. Univ. Sangio Kyot. 16, 1-15 (1985)]. The author in fact succeeds in proving that for each \(\gamma \geq 16\) and \(g \geq 6 \gamma + 4\), there is a \(\gamma\)-hyperelliptic symmetric numerical semigroup of genus \(g\) which is not realized as a Weierstrass semigroup.