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Weierstrass points and double coverings of curves. With application: Symmetric numerical semigroups which cannot be realized as Weierstrass semigroups. (English) Zbl 0838.14025
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 T. Kato, J. Komeda and 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 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 H. Ishida, 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.
Reviewer: R.Horiuchi (Kyoto)

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group
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References:
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