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Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility
\textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic
properties of the Riemann surfaces they are embedded in. In general, it is not
easy to describe algebraic properties of the surface of the embedding starting
from the combinatorial properties of an embedded dessin. However, this task
becomes easier if the dessin has a large automorphism group.
In this paper we consider a special type of dessins, so-called \textit{Wada
dessins}. Their underlying graph illustrates the incidence structure of finite
projective spaces \PR{m}{n}. Usually, the automorphism group of these dessins
is a cyclic \textit{Singer group} permuting transitively the
vertices. However, in some cases, a second group of automorphisms
exists. It is a cyclic group generated by the \textit{Frobenius automorphism}.
We show under what conditions is a group of automorphisms acting
freely on the edges of the considered dessins.Comment: 23 page
LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies
We extend the classical LR characterization of chirotopes of finite planar
families of points to chirotopes of finite planar families of pairwise disjoint
convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a
chirotope of finite planar families of pairwise disjoint convex bodies if and
only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the
set of 3-subsets of J is a chirotope of finite planar families of pairwise
disjoint convex bodies. Our main tool is the polarity map, i.e., the map that
assigns to a convex body the set of lines missing its interior, from which we
derive the key notion of arrangements of double pseudolines, introduced for the
first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio
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