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Special pairs in the generating subset of the Weierstrass semigroup at a pair. (English) Zbl 1019.14016
Summary: We discuss the structure of the Weierstrass semigroup at a pair of points on an algebraic curve. It is known [see M. Homma, Arch. Math. 67, 337-348 (1996; Zbl 0869.14015) and S. J. Kim and J. Komeda, Bol. Soc., Bras. Mat., Nova Sér. 32, No. 2, 149-157 (2001; Zbl 1077.14534)] that the Weierstrass semigroup at a pair $$(P,Q)$$ contains the unique generating subset $$\Gamma(P,Q)$$. We find some characterizations of the elements of $$\Gamma(P,Q)$$ and prove that, for any point $$P$$ on a curve, $$\Gamma(P,Q)$$ consists of only maximal elements for all except for finitely many points $$Q\neq P$$ on the given curve. Also we obtain more results concerning special and non-special pairs.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 30F10 Compact Riemann surfaces and uniformization 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14G50 Applications to coding theory and cryptography of arithmetic geometry
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