1,628 research outputs found
A Tropical Toolkit
We give an introduction to Tropical Geometry and prove some results in
Tropical Intersection Theory. The first part of this paper is an introduction
to tropical geometry aimed at researchers in Algebraic Geometry from the point
of view of degenerations of varieties using projective not-necessarily-normal
toric varieties. The second part is a foundational account of tropical
intersection theory with proofs of some new theorems relating it to classical
intersection theory.
Revised version includes many corrections, more examples, and improved
exposition.Comment: 38 page
Lifting Tropical Curves in Space and Linear Systems on Graphs
Tropicalization is a procedure for associating a polyhedral complex in
Euclidean space to a subvariety of an algebraic torus. We study the question of
which graphs arise from tropicalizing algebraic curves. By using Baker's
specialization of linear systems from curves to graphs, we are able to give a
necessary condition for a balanced weighted graph to be the tropicalization of
a curve. Our condition reproduces a generalization of Speyer's well-spacedness
condition and also gives new conditions. In addition, it suggests a new
combinatorial structure on tropicalizations of algebraic curves.Comment: substantial revisio
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