The uniqueness of Weierstrass points with semigroup ⟨a; b⟩ and related semigroups

Assume a and b = na + r with n ≥ 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 Ca;b-curve. In case r ≠ a − 1 and b ≠ a + 1 we prove C has no other point Q ≠ 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 Ca;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.

Publication year:2019

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Authors from:Higher Education

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