Connectedness and combinatorial interplay in the moduli space of line arrangements
Abstract
This paper aims to undertake an exploration of the behavior of the moduli space of line arrangements while establishing its combinatorial interplay with the incidence structure of the arrangement. In the first part, we investigate combinatorial classes of arrangements whose moduli space is connected. We unify the classes of simple and inductively connected arrangements appearing in the literature. Then, we introduce the notion of arrangements with a rigid pencil form. It ensures the connectedness of the moduli space and is less restrictive that the class of $C_3$ arrangements of simple type. In the last part, we obtain a combinatorial upper bound on the number of connected components of the moduli space. Then, we exhibit examples with an arbitrarily large number of connected components for which this upper bound is sharp.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.00322
 arXiv:
 arXiv:2309.00322
 Bibcode:
 2023arXiv230900322G
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 14H10;
 14N20;
 51A45;
 14N10
 EPrint:
 20 pages, 5 figures. Minor revisions from v.1