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On the Complexity of the k-Level in Arrangements of Pseudoplanes
A classical open problem in combinatorial geometry is to obtain tight
asymptotic bounds on the maximum number of k-level vertices in an arrangement
of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes
passing below them). This is a dual version of the k-set problem, which, in a
primal setting, seeks bounds for the maximum number of k-sets determined by n
points in d dimensions, where a k-set is a subset of size k that can be
separated from its complement by a hyperplane. The k-set problem is still wide
open even in the plane, with a substantial gap between the best known upper and
lower bounds. The gap gets larger as the dimension grows. In three dimensions,
the best known upper bound is O(nk^(3/2)).
In its dual version, the problem can be generalized by replacing hyperplanes
by other families of surfaces (or curves in the planes). Reasonably sharp
bounds have been obtained for curves in the plane, but the known upper bounds
are rather weak for more general surfaces, already in three dimensions, except
for the case of triangles. The best known general bound, due to Chan is
O(n^2.997), for families of surfaces that satisfy certain (fairly weak)
properties.
In this paper we consider the case of pseudoplanes in 3 dimensions (defined
in detail in the introduction), and establish the upper bound O(nk^(5/3)) for
the number of k-level vertices in an arrangement of n pseudoplanes. The bound
is obtained by establishing suitable (and nontrivial) extensions of dual
versions of classical tools that have been used in studying the primal k-set
problem, such as the Lova'sz Lemma and the Crossing Lemma.Comment: 23 pages, 13 figure
Tropicalization of classical moduli spaces
The image of the complement of a hyperplane arrangement under a monomial map
can be tropicalized combinatorially using matroid theory. We apply this to
classical moduli spaces that are associated with complex reflection
arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa
quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our
primary example is the Burkhardt quartic, whose tropicalization is a
3-dimensional fan in 39-dimensional space. This effectuates a synthesis of
concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page
Flipping Cubical Meshes
We define and examine flip operations for quadrilateral and hexahedral
meshes, similar to the flipping transformations previously used in triangular
and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th
International Meshing Roundtable. This version removes some unwanted
paragraph breaks from the previous version; the text is unchange
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