On curves with all Weierstrass points of maximal weight
Let \(X\) be a smooth irreducible projective curve of genus \(g \geq 2\) defined over the complex field. A point \(P \in X\) is called a Weierstrass point of \(X\) if the weight \(w(P)\) of \(P\) is positive, where \(w(P)=\sum{(\ell_j-j)}\) and \(\ell_1, \ell_2,\dots,\ell_g\) is the Weierstrass gap sequence of \(P\). The number of all Weierstrass points of \(X\) counted with their weight is \(g^3-g\). Moreover the integer \((g-1)g/2\) is an upper bound for \(w(P)\) and if the equality holds for some point on \(X\), then the curve is hyperelliptic and all Weierstrass points of \(X\) have maximal weight. If \(X\) is a non-hyperelliptic curve then \textit{T. Kato} [Math. Ann. 239, 141--147 (1979; Zbl 0401.30037)] obtained upper bounds for the weight of each Weierstrass point of \(X\). In the paper under review, the authors consider non-hyperelliptic curves of genus \(g\) on which the weights of all Weierstrass points attain the Kato's bounds. They start by showing that the genus of a such curve \(X\) is 3 or 5. The first case occurs only for the well-known Fermat quartic and the curve obtained by \textit{A. Kuribayashi} and \textit{K. Komiya} [Hiroshima Math. J. 7, 743--768 (1977; Zbl 0398.30035)]. In the case \(g=5\) the authors proved that \(X\) is the only curve of genus five with maximal number of automorphisms. To do this they studied the existence of elliptic involutions on \(X\).
scientific article; zbMATH DE number 5708653
scientific article