Some remarks on Humbert-Edge's curves
Let \(X \subset \mathbb P^4\) be a canonical curve of genus \(5\). According to \textit{W. L. Edge} [Proc. Camb. Philos. Soc. 47, 483--495 (1951; Zbl 0044.35503)], \(X\) is a \textit{Humbert curve} if \(X\) is the complete intersection of three diagonal quadrics \(Q_i = \sum_{j=0}^4 a_{ij} x_j^2\) after a linear coordinate change, because projections of such a curve into \(\mathbb P^3\) were discovered and studied by \textit{G. Humbert} in the 1890s [J. de l'Éc. Pol. 64, 123--149 (1894; JFM 25.1252.01)]. The authors prove some criteria for \(X \subset \mathbb P^4\) to be a Humbert curve. These conditions were stated by \textit{R. Varley} [Am. J. Math. 108, 931--951 (1986; Zbl 0604.14031)] and possibly known to Edge. Then they consider a generalization due to \textit{Edge} [loc. cit.], integral non-degenerate non-singular curves \(X_n \subset \mathbb P^n\) which are complete intersections of \(n-1\) diagonal quadrics \(Q_1, \dots, Q_{n-1}\), calling them \textit{Humbert-Edge curves}. They prove that a smooth connected curve \(X \subset \mathbb P^n\) of genus \(2^{n-2} (n-3)+1\) is a Humbert-Edge curve if and only if \(X\) admits an action of a group \(E \cong (\mathbb Z / 2 \mathbb Z)^n\) with generators \(\sigma_0, \dots, \sigma_n\) such that \(\sigma_0 \dots \sigma_n=1\) and the fixed sets of \(\sigma_i\) are pairwise disjoint with cardinality \(2^{n-1}\). This equivalence was proved by Carocca, González-Aguilera, Hidalgo and Rodríguez over the complex numbers [\textit{A. Carocca} et al., Isr. J. Math. 164, 165--192 (2008; Zbl 1151.30030)], but here the authors give an algebraic proof over an algebraic closed field of characteristic not equal to two. They also provide a partial equivariant decomposition of the Jacobian variety of \(X\) with respect to the action of \(E\).
scientific article; zbMATH DE number 7073896