A curve of genus 5 having 24 Weierstrass points of weight 5 (Q2634718)

Any bielliptic curve \(X\) of genus \(g\geq 5\) over the complex number field has only two types, \(I_g\) and \(II_g\), of Weierstrass points associated to one of the \(2g-2\) fixed points of a bi-elliptic involution and if \(g>6\) and \(0 < s \leq 2g-2\), \(s\neq 2g-3\), there is some \(X\) with exactly \(s\) fixed points of type \(II_g\) [\textit{J. Park}, Manuscripta Math. 95, 33--45 (1998; Zbl 0915.14020); \textit{E. Ballico} and \textit{S. J. Kim}, Indag. Math. New Ser. 9, No. 2, 155--159 (1998; Zbl 0930.14023); \textit{E. Ballico} and \textit{Del Centina}, Ann. Mat. Pura Appl. (4) 177, 293--313 (1999; Zbl 0997.14016)] and conversely if \(X\) has a Weiertrass point \(P\) of type \(I_g\) or \(II_g\) and either \(g>11\) or \(g=8\), then \(X\) is bielliptic and \(P\) is a fixed point of the bielliptic involution [\textit{T. Kato} Math. Ann 239, 141--147 (1979; Zbl 0461.30037); \textit{J. Komeda}, J. Reine Angew. Math. 341, 68--86 (1983; Zbl 0498.30053)]. The case \(g=5\) is much more complicated, because \(X\) may have \(>1\) bielliptic involutions (see [\textit{T. Kato} et al., Indag. Math. New Ser, 22, 141--147 (2011; Zbl 1259.14038)]). In the paper under review the main results is a characterization of genus \(5\) curves \(X\) with 24 Weierstrass points of type \(II_5\), i.e. with gap sequence \(\{1,2,3,5,9\}\) (it is the Wiman curve, it has \(3\) bielliptic involutions and these \(24\) points are the fixed points of one of the involutions). He also gives a criterion of \((X,P)\) implying that \(X\) is bielliptic with \(P\) a fixed point of an involution.