Special pairs in the generating subset of the Weierstrass semigroup at a pair. (English) Zbl 1019.14016

Kang, Eunju; Kim, Seon Jeong

Geom. Dedicata 99, 167-177 (2003).

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

Keywords:

generalized Weierstrass point; Weierstrass semigroup of a pair; Weierstrass semigroup of a point