Weights of Weierstrass points in double coverings of curves of genus one or two. (English) Zbl 0603.14014

Garcia, Arnaldo

Manuscr. Math. 55, 419-432 (1986).

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

Keywords:

weight of Weierstrass point; compact Riemann surface; sequence of non- gaps

References:

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