On Weierstrass semigroups of double covering of genus two curves (Q958187)

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 \((*)\). \textit{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 \textit{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.