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Weierstrass semigroups at inflection points. (English) Zbl 0869.14014
Let $$C$$ denote a nonsingular projective plane curve of degree $$d$$ over the complex numbers. Let $$P\in C$$ and let $$L$$ denote the tangent line to $$C$$ at $$P$$. Let $$C\cdot L$$ denote the divisor on $$C$$ cut out by $$L$$. The author first shows that if $$C\cdot L\geq (\lfloor d/2\rfloor +1)P$$, then $$P$$ is a Weierstrass point of $$C$$. The semigroup of Weierstrass non-gaps at $$P$$ when $$C\cdot L=dP$$ was determined by M. Coppens and T. Kato [Tsukuba J. Math. 18, No. 1, 119-129 (1994; Zbl 0819.14012)]. The author determines the semigroup of non-gaps at $$P$$ when $$C\cdot L=(d-1)P+Q$$ and when $$C\cdot L=(d-2)P+2Q$$, for some point $$Q\in C$$. When $$C\cdot L=(d-3)P+3Q$$, the author finds a set that contains the semigroup of nongaps. The proofs make use of previous results due to the author [S. J. Kim, Arch. Math. 62, No. 1, 73-82 (1994; Zbl 0815.14020)] concerning the Weierstrass semigroup of a pair of points on a curve.
##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences
##### Keywords:
inflection point; Weierstrass point; Weierstrass non-gaps