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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.
14H55 Riemann surfaces; Weierstrass points; gap sequences