Sectional monodromy groups of projective curves

Abstract

Fix a degree $d$ projective curve $X\subset {\mathbb{P}}^r$ over an algebraically closed field $K$. Let $U\subset ({\mathbb{P}}^r{)}^{\ast }$ be a dense open subvariety such that every hyperplane $H\in U$ intersects $X$ in $d$ smooth points. Varying $H\in U$ produces the monodromy action $\phi :{\pi }_1^{\text{eˊt}}(U)\to S_d$. Let $G_X:=\mathrm{im}(\phi )$. The permutation group $G_X$ is called the sectional monodromy group of $X$. In characteristic zero $G_X$ is always the full symmetric group, but sectional monodromy groups in characteristic $p$ can be smaller. For a large class of space curves ($r⩾3$) we classify all possibilities for the sectional monodromy group $G$ as well as the curves with $G_X=G$. We apply similar methods to study a particular family of rational curves in ${\mathbb{P}}^2$, which enables us to answer an old question about Galois groups of generic trinomials.

to appear in Journal of the London Mathematical Society