Tropical Skeletons

Abstract

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of X as the subset of the associated Berkovich space X an which collects all Shilov boundary points in the fibers of the Kajiwara-Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When X is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm-Katz.

Tropical geometry of genus two curves

“…This can be achieved by means of an extended Berkovich skeleton Σ(X ) coming from a semistable model of X with a horizontal divisor (i.e. the closure of a divisor of the generic fiber in the model) that is compatible with Γ [30,31]. Indeed, to each vertex v in Γ we assign the sum of the genera of all semistable vertices of Σ(X ) mapping to v under trop : Σ(X ) → Trop X .…”

Tropical Computations in polymake

“…Joining edges by omitting those of degree two gives the combinatorial skeleton of C, which is a planar graph with 2g − 2 vertices and 3g − 3 edges, where g is the genus, i.e., g = 3 in our case. The name comes about from its (loose) connection to the Berkovich skeleton of the analytification of a smooth complete curve; see [24]. The joined edges receive the sum of the lengths of the original edges of the curve, and this way we arrive at a metric graph.…”

Equations and Tropicalization of Enriques Surfaces

“…Moreover all multiplicities of top degree polyhedra are equal to one, hence the multiplicity at each point is equal to one by semicontinuity, see [23,Lemma 3.3.6]. Therefore π has a section trop(S) → S an whose image is equal to a skeleton S(S , H) of a suitable semistable model (S , H) of S, see [17,Remark 9.12]. The skeleton S(S , H) is a proper strong deformation retract of S an by [18, §4.9].…”