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On the weight of higher order Weierstrass points. (English) Zbl 0592.14019
For a divisor D on a complete non-singular curve of genus at least 2 over an algebraically closed field the notion of D-weight, \(w_ D(P)\), of a point P has been defined [D. Laksov, Astérisque 87/88, 221-247 (1981; Zbl 0489.14007)]. P is said to be a D-Weierstrass point if its D- weight is positive. In the paper under review the following results are established:
For a divisor D of degree at least 2g-1 on C \((a)\quad w_ D(P)\leq g(g+1)\) with equality if and only if C is hyperelliptic, P is a K- Weierstrass point, and D is linearly equivalent to \(K+(d-2g+2)P.\)
For non-hyperelliptic C, \((b)\quad w_ D(P)\leq k(g)+g,\) (see below) with the maximum achieved for every value of the degree and \(g\geq 3\), and \((c)\quad if\) equality occurs in (b), \(w_ K(P)=k(g).\)
Here k(g) is g(g-1)/3 if g is 3,4,6,7, or 9, and \((g^ 2-5g+10)/2\) if g is 5,8, or at least 10. - This extends a result of T. Kato [Math. Ann. 239, 141-147 (1979; Zbl 0401.30037)]. A counterexample to a conjecture of Duma is given, and conditions for its validity discussed.
Reviewer: H.H.Martens

14H55 Riemann surfaces; Weierstrass points; gap sequences
30F10 Compact Riemann surfaces and uniformization
14H30 Coverings of curves, fundamental group
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