On the monodromy of Weierstrass points
Suppose given two complementary subspaces \({\mathbb{P}}^{m-1}\) of \({\mathbb{P}}^{g-1}\) with \(g=2m\geq 4\), let \(\Gamma \subset {\mathbb{P}}^{m-1}\) be a rational normal curve, and let S be the rational scroll (surface) consisting of the lines joining the corresponding points on the two copies of \(\Gamma\). Consider all canonically embedded curves \(C\subset {\mathbb{P}}^{g-1}\) of genus \(g\) lying on S. Then the linear system \(X=| C|\) on S has dimension \(2g+7\). Let I be the set of triples (C,z,H) with \(C\in X\), \(z\in S\), and H a hyperplane in \({\mathbb{P}}^{g-1}\) such that \(m_ z(C.H)\geq g\). The author proves there is exactly one component Y of I of dimension \(2g+7=\dim X\) mapping onto X. This can be phrased as saying that this system \(Y\to X\) of curves with a certain Weierstrass point has a transitive monodromy group, i.e. if \(L\supset K(X)\) is the normal closure of K(Y)\(\supset K(X)\), the group Gal(L/K(X)) acts transitively on the set of Weierstrass points \(z\in C\). For \(g=4\) this group turns out to be the full symmetric group. It may be that the same is true for any \(g=2m>4\).
scientific article; zbMATH DE number 3977143