Béranger Seguin

Contact me: beranger[dot]seguin[at]ens[dot]fr

I am a third year PhD student supervised by Pierre Dèbes and Ariane Mézard.

My current work is centered around the use of geometric methods for number-theoretical purposes. I have spent the last few years working on the regular inverse Galois problem and, to this end, on the geometry and arithmetic of Hurwitz spaces, which are moduli spaces of branched covers of \(\mathbb{P}^1\).

More precisely, I study the irreducible components of these spaces using combinatorial, topological and arithmetical approaches. The links between these points of view are, in my eyes, as mysterious as they are fascinating, and it is with joy that I humbly try to contribute to their elucidation.

In parallel, I like to secretly travel to other mathematical landscapes:

logic and proof theory (I am interested in formalization programs)
higher category theory
non-commutative algebra.

I also happen to like music (especially jazz) a lot, and I’m always glad to discuss music or play with people.

Fields of Definition of Components of Hurwitz Spaces (preprint)

Béranger Seguin

2023

28 pages, 3 figures.

Abs arXiv Bib PDF

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.
@misc{fielddefcomp, author = {Seguin, Béranger}, title = {Fields of Definition of Components of Hurwitz Spaces (preprint)}, year = {2023}, eprint = {arXiv:2303.05903}, note = {28 pages, 3 figures.} }
The Geometry of Rings of Components of Hurwitz Spaces (preprint)

Béranger Seguin

2022

52 pages.

Abs arXiv Bib PDF

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 (preprint)}, year = {2022}, eprint = {arXiv:2210.12793}, note = {52 pages.} }