Joseph Rabinoff

Nonarchimedean geometry, tropicalization, and metrics on curves

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 an isometry on finite subgraphs. Other applications include generalizations of Speyers well-spacedness condition and the Katz-Markwig-Markwig results on tropical j-invariants.

Lifting harmonic morphisms II: Tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.

Uniform bounds for the number of rational points on curves of small Mordell–Weil rank

Let

Non-Archimedean and tropical theta functions

X

Hybrid grids and the Homing Robot

be a curve of genus

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

g\geq 2

Tropical analytic geometry, Newton polygons, and tropical intersections

over a number field

Skeletons and tropicalizations

F

On the structure of nonarchimedean analytic curves

of degree

The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves

d = [F:Q]