1,246 research outputs found
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
The injectivity radius of hyperbolic surfaces and some Morse functions over moduli spaces
This article is devoted to the variational study of two functions defined
over some Teichmueller spaces of hyperbolic surfaces. One is the systole of
geodesic loops based at some fixed point, and the other one is the systole of
arcs.\par For each of them we determine all the critical points. It appears
that the systole of arcs is a topological Morse function, whereas the systole
of geodesic loops have some degenerate critical points. However, these
degenerate critical points are in some sense the obvious one, and they do not
interfere in the variational study of the function.\par At a nondegenerate
critical point, the systolic curves (arcs or loops depending on the function
involved) decompose the surface into regular polygons. This enables a complete
classification of these points, and some explicit computations. In particular
we determine the global maxima of these functions. This generalizes optimal
inequalities due to Bavard and Deblois. We also observe that there is only one
local maximum, this was already proved in some cases by Deblois.\par Our
approach is based on the geometric Vorono\''i theory developed by Bavard. To
use this variational framework, one has to show that the length functions (of
arcs or loops) have positive definite Hessians with respect to some system of
coordinates for the Teichm\''uller space. Following a previous work, we choose
Bonahon's shearing coordinates, and we compute explicitly the Hessian of the
length functions of geodesic loops. Then we use a characterization of the
nondegenerate critical points due to Akrout.Comment: Preliminary version, to be improved shortly.The author is fully
supported by the fund FIRB 2010 (RBFR10GHHH003
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