587 research outputs found
Stable Intersections of Tropical Varieties
We give several characterizations of stable intersections of tropical cycles
and establish their fundamental properties. We prove that the stable
intersection of two tropical varieties is the tropicalization of the
intersection of the classical varieties after a generic rescaling. A proof of
Bernstein's theorem follows from this. We prove that the tropical intersection
ring of tropical cycle fans is isomorphic to McMullen's polytope algebra. It
follows that every tropical cycle fan is a linear combination of pure powers of
tropical hypersurfaces, which are always realizable. We prove that every stable
intersection of constant coefficient tropical varieties defined by prime ideals
is connected through codimension one. We also give an example of a realizable
tropical variety that is connected through codimension one but whose stable
intersection with a hyperplane is not.Comment: Revised version, to appear in Journal of Algebraic Combinatoric
The space of tropically collinear points is shellable
The space T_{d,n} of n tropically collinear points in a fixed tropical
projective space TP^{d-1} is equivalent to the tropicalization of the
determinantal variety of matrices of rank at most 2, which consists of real d x
n matrices of tropical or Kapranov rank at most 2, modulo projective
equivalence of columns. We show that it is equal to the image of the moduli
space M_{0,n}(TP^{d-1},1) of n-marked tropical lines in TP^{d-1} under the
evaluation map. Thus we derive a natural simplicial fan structure for T_{d,n}
using a simplicial fan structure of M_{0,n}(TP^{d-1},1) which coincides with
that of the space of phylogenetic trees on d+n taxa. The space of phylogenetic
trees has been shown to be shellable by Trappmann and Ziegler. Using a similar
method, we show that T_{d,n} is shellable with our simplicial fan structure and
compute the homology of the link of the origin. The shellability of T_{d,n} has
been conjectured by Develin in 2005.Comment: final version, minor revision, 15 page
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