1,094 research outputs found
Tropical Severi Varieties
We study the tropicalizations of Severi varieties, which we call tropical
Severi varieties. In this paper, we give a partial answer to the following
question, ``describe the tropical Severi varieties explicitly.'' We obtain a
description of tropical Severi varieties in terms of regular subdivisions of
polygons. As an intermediate step, we construct explicit parameter spaces of
curves. These parameter spaces are much simpler objects than the corresponding
Severi variety and they are closely related to flat degenerations of the Severi
variety, which in turn describes the tropical Severi variety. As an
application, we understand G.Mikhalkin's correspondence theorem for the degrees
of Severi varieties in terms of tropical intersection theory. In particular,
this provides a proof of the independence of point-configurations in the
enumeration of tropical nodal curves.Comment: 25 pages, Final version accepted to Portugal. Mat
Convex lattice polygons of fixed area with perimeter dependent weights
We study fully convex polygons with a given area, and variable perimeter
length on square and hexagonal lattices. We attach a weight t^m to a convex
polygon of perimeter m and show that the sum of weights of all polygons with a
fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less
than a critical threshold t_c, where K is a t-dependent constant, and
theta_{conv} is a critical exponent which does not change with t. We find
theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice.
The reason for this unexpected non-universality of theta_{conv} is traced to
existence of sharp corners in the asymptotic shape of these polygons.Comment: 8 pages, 5 figures, revtex
Brief introduction to tropical geometry
The paper consists of lecture notes for a mini-course given by the authors at
the G\"okova Geometry \& Topology conference in May 2014. We start the
exposition with tropical curves in the plane and their applications to problems
in classical enumerative geometry, and continue with a look at more general
tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise
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