Preprints and publications

2023

1. 

   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.}
   }

2022

1. 

   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.}
   }

2. 

   Study of a division-like property (preprint)

   Robin Khanfir, and Béranger Seguin

   2022

   15 pages. Unrelated to PhD work.

   Abs arXiv Bib PDF

   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.

   @misc{fadelian,
     author = {Khanfir, Robin and Seguin, Béranger},
     title = {Study of a division-like property (preprint)},
     year = {2022},
     eprint = {arXiv:2210.12793},
     note = {15 pages. Unrelated to PhD work.}
   }