3,630 research outputs found

    Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility

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    \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} Σℓ\Sigma_\ell permuting transitively the vertices. However, in some cases, a second group of automorphisms Φf\Phi_f exists. It is a cyclic group generated by the \textit{Frobenius automorphism}. We show under what conditions Φf\Phi_f 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

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    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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