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On Weierstrass semigroups of double covering of genus two curves. (English) Zbl 1161.14023
The authors investigate the Weierstrass property of (numerical) semigroups of the form $$H=2\tilde H\cup\{u_2<u_1\}\cup\{2g+i:i\in{\mathbb N}_0\}$$ (*), where $$\tilde H$$ is a semigroup of genus two, $$u_2,u_1$$ odd numbers and $$g$$ the genus of $$H$$. Examples of such semigroups are the Weierstrass semigroups at totally ramified points of double covering of genus two curves. The main problem is concerning the Weierstrass property of $$(*)$$. A. Garcia [Manuscr. Math. 55, 419–432 (1986; Zbl 0824.14033)] computed all the possibilities for the semigroups $$(*)$$, where $$\{u_2,u_1\}$$ may be $$\{2g-3, 2g-1\}\, (*1)$$, $$\{2g-7,2g-1\}\, (*_2)$$, $$\{2g-5,2g-3\}\, (*_3)$$ and $$\{2g-5,2g-1\}\, (*_4)$$. The case $$4\in H$$ is always Weierstrass by a result of J. Komeda [J. Reine Angew. Math. 341, 68–86 (1983; Zbl 0498.30053)]. If $$6\in H$$, Garcia showed that the cases $$(*_1),\ldots,(*_3)$$ are Weierstrass and $$(*_4$$) is so for $$g\equiv 1\pmod{3}$$. In this paper it is proved that in fact all the semigroups listed above are Weierstrass for $$g$$ large enough.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences
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##### References:
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