Abstract
Assume a and \(b=na+r\) with \(n \ge 1\) and \(0<r<a\) are relatively prime integers. In case C is a smooth curve and P is a point on C with Weierstrass semigroup equal to \(<a;b>\) then C is called a \(C_{a;b}\)-curve. In case \(r \ne a-1\) and \(b \ne a+1\) we prove C has no other point \(Q \ne P\) having Weierstrass semigroup equal to \(<a;b>\), in which case we say that the Weierstrass semigroup \(<a;b>\) occurs at most once. The curve \(C_{a;b}\) has genus \((a-1)(b-1)/2\) and the result is generalized to genus \(g<(a-1)(b-1)/2\). We obtain a lower bound on g (sharp in many cases) such that all Weierstrass semigroups of genus g containing \(<a;b>\) occur at most once.
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Communicated by Daniel Greb.
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Coppens, M. The uniqueness of Weierstrass points with semigroup \(\langle a;b\rangle \) and related semigroups. Abh. Math. Semin. Univ. Hambg. 89, 1–16 (2019). https://doi.org/10.1007/s12188-019-00201-y
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DOI: https://doi.org/10.1007/s12188-019-00201-y