We study a weak divisibility property for noncommutative rings: a nontrivial ring is
fadelian if for all nonzero a and x there exist b, c such that x = ab + ca. We prove properties of
fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian
and non-Ore examples.
@article{fadelian,author={Khanfir, Robin and Seguin, Béranger},title={Study of a division-like property},year={2024},doi={10.1142/S0219498825502214},journal={Journal of Algebra and its Applications},}
2023
Geometry and Arithmetic of Components of Hurwitz Spaces
Hurwitz spaces are moduli spaces that classify ramified covers of the projective line on
which a fixed group G acts. Their geometric and arithmetic properties are related to
number theoretical questions, particularly the inverse Galois problem. In this thesis,
we study the connected components of these spaces. Firstly, we prove results concerning the asymptotical behaviour of the count of connected components of Hurwitz
spaces as the number of branch points of the covers they classify grows. Secondly, we
establish stability results for fields of definitions of connected components of Hurwitz
spaces under the gluing operation. These results relate topological and arithmetical
properties of covers. Three expository chapters, devoid of original statements, present
the various objects. In an appendix, we summarize the thesis for the general public..
@misc{these,author={Seguin, Béranger},title={Geometry and Arithmetic of Components of Hurwitz Spaces},year={2023},note={PhD thesis.}}
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. Applications are given.
@misc{fielddefcomp,author={Seguin, Béranger},title={Fields of Definition of Components of Hurwitz Spaces},year={2023},eprint={arXiv:2303.05903},note={Preprint. 28 pages, 3 figures.}}
2022
The Geometry of Rings of Components of Hurwitz Spaces
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.
@misc{geomringcomp,author={Seguin, Béranger},title={The Geometry of Rings of Components of Hurwitz Spaces},year={2022},eprint={arXiv:2210.12793},note={Preprint. 52 pages.}}
