Abstract
Suppose that X is a compact Riemann surface of genus g ≥ 2, while σ is an automorphism of X of order n, and g* is the genus of the quotient surface X* = X/〈σ〉. In 1951 Schöneberg obtained a sufficient condition for a fixed point P ∈ X of σ to be a Weierstrass point of X. Namely, he showed that P is a Weierstrass point of X if g* ≠ [g/n], where [x] is the integral part of x. Somewhat later Lewittes proved the following theorem, equivalent to Schöneberg’s theorem: If a nontrivial automorphism σ fixes more than four points of X then all of them are Weierstrass points.
These assertions are connected with the notion of a regular covering. We generalize the Lewittes theorem to the case of nonregular coverings and obtain some related corollaries.
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Original Russian Text Copyright © 2014 Limonov M.P.
The author was partially supported by the Laboratory of Quantum Topology at Chelyabinsk State University (Grant 14.Z50.31.0020 of the Government of the Russian Federation) and the Russian Foundation for Basic Research (Grant 12-01-00210).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 6, pp. 1328–1333, November–December, 2014.
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Limonov, M.P. On a generalization of the Lewittes theorem on Weierstrass points. Sib Math J 55, 1084–1088 (2014). https://doi.org/10.1134/S003744661406010X
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DOI: https://doi.org/10.1134/S003744661406010X