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 \mathbbQ of small dimension (6 and 4, respectively) are shown to exist.

2022

The Geometry of Rings of Components of Hurwitz Spaces (preprint)

We study a variant of the ring of components of Hurwitz moduli spaces for covers, introduced by Ellenberg, Venkatesh and Westerland. For G-covers of the projective line, the ring of components is a commutative algebra of finite type. We study it using tools from algebraic geometry. We relate the geometry of its spectrum to group-theoretical properties and combinatorial aspects of Galois covers. When G is a symmetric group, we fully describe the geometric points of the spectrum.

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a, x there exist b, c such that x=ab+ca. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. Some of these results are formalized using the Lean proof assistant.