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Weights of Weierstrass points in double coverings of curves of genus one or two. (English) Zbl 0603.14014
Let X be a compact Riemann surface of genus $$g.$$ A Weierstrass point $$P\in X$$ has non-zero weight w(P) which is defined by $$w(P)=g(3g+1)- \sum^{g}_{j=1}m_ j$$ where $$\{m_ j\}$$ is the sequence of non-gaps at P. The authors proves: if $$g\geq 11$$, and X is non-hyperelliptic, then X is a double covering of a surface of genus $$1$$ if and only if there exists $$P\in X$$ with $$g^ 2-g>2w(P)\geq g^ 2-5g+6$$; and a similar result with genus $$1$$ replaced by genus $$2.$$ This extends the results and uses the methods of T. Kato [Math. Ann. 239, 141-147 (1979; Zbl 0401.30037)]. The techniques are standard in this area and use extensively the fact that the non-gaps form a semi-group.
An interesting key lemma: If X is non-hyperelliptic, then for $$P\in X$$, $$m_ j\geq 2j+1$$ for $$1\leq j\leq g-2$$, $$m_{g-1}\leq 2g-2$$ and $$m_ g=2g$$.
Reviewer: C.Maclachlan

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H45 Special algebraic curves and curves of low genus 30F10 Compact Riemann surfaces and uniformization
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##### References:
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