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On Perez del Pozo’s lower bound of Weierstrass weight. (English) Zbl 1403.14066
Let \(C\) be a smooth projective curve over the complex number field with genus at least 2 and \(\sigma\) an automorphism on \(C\) such that the curve \(C/<\sigma>\) has genus 0. In [Arch. Math. 86, No. 1, 50–55 (2006; Zbl 1093.14045)] A. L. Perez del Pozo obtained a lower bound, \(w\), of the Weierstrass weights of fixed points of \(\sigma\).In the present paper the authors find necessary and sufficient conditiomns for when this lower bound \(w\) occurs.More precisely, they show that if the number of fixed points is at least 3 and \(P\) is a fixed point of \(\sigma\) with Weierstrass weight \(w(P) = w\), the lower bound, then for the cyclic covering \(\pi\) : \(C\)\( \rightarrow\) \(C/<\sigma>\) all the ramification points are total ramifications. Furthermore, they give an algorithm to compute the gap sequences of the ramification points of \(\pi\). In addition, for the case of a unique fixed point, they give a classification of those curves \(C\) where the Weierstrass weight of the unique fixed point is 1, 2 or 3. [K. Yoshida, Tsukuba J. Math. 17, No. 1, 221–249 (1993; Zbl 0790.30030)].
MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H37 Automorphisms of curves
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