Fields of Definition of Components of Hurwitz Spaces

For a fixed finite group $G$, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched $G$-covers of the projective line. The main focus is on determining whether components obtained by “gluing” two other components, both defined over a number field $K$, are also defined over $K$. The article presents a list of situations in which a positive answer is obtained. As an application, when $G$ is a semi-direct product of symmetric groups or the Mathieu group $M_{23}$, components defined over $\mathbb{Q}$ of small dimension ($6$ and $4$, respectively) are shown to exist.