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The Weierstrass semigroup of a pair of points on a curve. (English) Zbl 0869.14015
Let \(X\) be a smooth, projective curve of genus \(g\geq 2\) over an algebraically closed field of arbitrary characteristic. Let \(P\) and \(Q\) be points on \(X\). The Weierstrass semigroup \(H(P,Q)\) of the pair \((P,Q)\) is defined to be the set of ordered pairs of non-negative integers \((\alpha,\beta) \in\mathbb{N}_0 \times\mathbb{N}_0\) such that there is a meromorphic function on \(X\) with polar divisor \(\alpha P+ \beta Q\). The gap set of the pair \((P,Q)\) is the complement of \(H(P,Q)\) in \(\mathbb{N}_0 \times \mathbb{N}_0\). S. J. Kim [Arch. Math. 62, No. 1, 73-82 (1994; Zbl 0815.14020)] gave upper and lower bounds for the cardinality of the gap set (assuming that the characteristic was zero). The author generalizes these bounds to positive characteristic, gives a geometric condition on certain osculating planes at \(P\) and \(Q\) that insures that the lower bound is achieved (if \(X\) is non-hyperelliptic), and shows that the lower bound is met for general (distinct) points \(P\) and \(Q\) on a non-hyperelliptic curve of genus 3 or 4. One should expect this lower bound to be met for a general pair of points for all \(g\), since this is the case in characteristic 0, as is shown here and as was stated in an exercise by E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris in their book: “Geometry of algebraic curves. I” (New York 1985; Zbl 0559.14017) in which Weierstrass pairs first appeared.
Let \(l_1, \dots, l_g\) (resp. \(l_1', \dots, l_g')\) denote the Weierstrass gaps at \(P\) (resp. at \(Q)\). S. J. Kim also showed that if \(l_i\) is a Weierstrass gap at \(P\), then \(\min \{\beta |(l_i, \beta) \in H(P,Q)\}\) is a gap \(l_{\sigma(i)}'\) at \(Q\). If \(r(\sigma)\) denotes the cardinality of the set of ordered pairs \((i,j)\) with \(i<j\) such that \(\sigma (i) >\sigma (j)\), then the author shows that the cardinality of the gap set of the pair \((P,Q)\), where \(P\) and \(Q\) are distinct, is given by \(\sum^g_{i=1} l_i+ \sum^g_{i=1} l_i'- r(\sigma)\).

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
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