224 research outputs found
Geometry of tropical moduli spaces and linkage of graphs
We prove the following "linkage" theorem: two p-regular graphs of the same
genus can be obtained from one another by a finite alternating sequence of
one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the
linkage theorem to prove that various moduli spaces of tropical curves are
connected through codimension one.Comment: Final version incorporating the referees correction
Algebraic and tropical curves: comparing their moduli spaces
We construct the moduli space for equivalence classes of n-pointed tropical
curves of genus g, together with its compactification given by weighted
tropical curves, and establish some of its basic topological properties. We
compare it to the moduli spaces of smooth and stable algebraic curves, from the
combinatorial, the topological, and the Teichm\"uller point of view. The paper
is written in an expository style, and it generalizes some results contained in
sections 4-6 of arXiv:1001.2815v3.Comment: Changes in numbering to match the published versio
Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles
We study rational double Hurwitz cycles, i.e. loci of marked rational stable
curves admitting a map to the projective line with assigned ramification
profiles over two fixed branch points. Generalizing the phenomenon observed for
double Hurwitz numbers, such cycles are piecewise polynomial in the entries of
the special ramification; the chambers of polynomiality and wall crossings have
an explicit and "modular" description. A main goal of this paper is to
simultaneously carry out this investigation for the corresponding objects in
tropical geometry, underlining a precise combinatorial duality between
classical and tropical Hurwitz theory
Tropical covers of curves and their moduli spaces
We define the tropical moduli space of covers of a tropical line in the plane
as weighted abstract polyhedral complex, and the tropical branch map recording
the images of the simple ramifications. Our main result is the invariance of
the degree of the branch map, which enables us to give a tropical
intersection-theoretic definition of tropical triple Hurwitz numbers. We show
that our intersection-theoretic definition coincides with the one given by
Bertrand, Brugall\'e and Mikhalkin in the article "Tropical Open Hurwitz
numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus we
provide a tropical intersection-theoretic justification for the multiplicities
with which a tropical cover has to be counted. Our method of proof is to
establish a local duality between our tropical moduli spaces and certain moduli
spaces of relative stable maps to the projective line.Comment: 24 pages, 10 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
Generic singular configurations of linkages
We study the topological and differentiable singularities of the
configuration space C(\Gamma) of a mechanical linkage \Gamma in d-dimensional
Euclidean space, defining an inductive sufficient condition to determine when a
configuration is singular. We show that this condition holds for generic
singularities, provide a mechanical interpretation, and give an example of a
type of mechanism for which this criterion identifies all singularities
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