Weights of Weierstrass points in double coverings of curves of genus one or two (Q1082390)

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 \textit{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 \(m_ g=2g\).