Nonarchimedean geometry, tropicalization, and metrics on curves

Abstract

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to isometries on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the KatzMarkwig-Markwig results on tropical j-invariants.

A tropical approach to nonarchimedean Arakelov geometry

“…This method to compute tropicalizations is due to [BPR13,BPR16] and we will refer to these papers for details of the following construction. Skeleta are discussed in [BPR13] and we refer to [BPR13, Corollary 4.23] for existence and uniqueness of the minimal skeleton S(W ) of the smooth curve W .…”

Moduli of tropical plane curves

“…The genus g = g(C) is the number of interior lattice points of P. Each bounded edge of C has a well-defined lattice length. The curve C contains a subdivision of a metric graph of genus g with vertices of valency ≥ 3 as in [5], and this subdivision is unique for g ≥ 2. The underlying graph G is planar and has g distinguished cycles, one for each interior lattice point of P. We call G the skeleton of C. It is the smallest subspace of C to which C admits a deformation retract.…”

Limits of tropicalizations

“…Tropicalizations of more general toric varieties were studied implicitly by Thuillier, as skeletons of analytifications [Thu07], and explicitly by Kajiwara [Kaj08], before being applied to the development of explicit relations between tropical and nonarchimedean analytic geometry in [Pay09a,BPR11,Rab12], to which we refer the reader for further details. The tropicalization of the affine toric variety U σ is the space of monoid homomorphisms…”