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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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