Sectional monodromy groups of projective curves

TL;DR

This paper classifies the possible monodromy groups of projective curves in various characteristics and applies these results to determine Galois groups of certain rational trinomials.

Contribution

It provides a classification of sectional monodromy groups for a broad class of space curves and explores their implications for Galois groups of polynomials.

Findings

In characteristic zero, the monodromy group is always the full symmetric group.

In positive characteristic, the monodromy group can be smaller and is classified for certain space curves.

The study answers an old question about Galois groups of generic trinomials.

Abstract

Fix a degree $d$ projective curve $X \subset \mathbb{P}^r$ over an algebraically closed field $K$. Let $U \subset (\mathbb{P}^r)^*$ 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 $\varphi: \pi_1^{\text{\'et}}(U) \to S_d$. Let $G_X := \mathrm{im}(\varphi)$. 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 ($r \geqslant 3$) 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.