The regular inverse Galois problem over fields avoiding a curve

Abstract

Let $k_0$ be a field of characteristic $0$, and let $C/k_0$ be a smooth geometrically integral curve of positive genus with $C(k_0)=\varnothing$. We prove that the regular inverse Galois problem is solvable over every field existentially closed in the class of $k_0$-fields avoiding $C$, including the curve-excluding fields of Johnson and the author. To this end, we show that for every finite group $G$, there is a parameter variety over $\mathbb{Q}$ whose function field admits a regular realization of $G$ without adding points to $C$. Finally, we establish $\mathrm{RIGP}$ over the variety-excluding fields of Szachniewicz and the author, when the excluded variety is a locally closed subvariety of an abelian variety.