Numerical tropical line bundles and toric b-divisors

TL;DR

This paper explores the connection between tropical line bundles on compactifications of varieties and toric b-divisors, establishing a correspondence that generalizes Baker's specialization to higher dimensions.

Contribution

It constructs an injective map linking numerical tropical line bundles to toric b-divisors and characterizes the tropical nef cone in terms of b-Cartier and nef divisors.

Findings

Established a bijection between tropical nef cones and b-Cartier, tropically nef b-divisors.

Clarified the kernel of the line bundle to tropical line bundle map, revealing lost moduli.

Extended Baker's specialization from curves to higher-dimensional varieties.

Abstract

We study the relationship between line bundles on tropical compactifications of a very affine variety $Y$ and toric b-divisors on the associated tropical variety ${\rm Trop}(Y)$. By focusing on numerical equivalence classes, we construct a natural injective map from the group of numerical tropical line bundles on $Y$ to the space of toric b-divisors modulo linear equivalence. Moreover, we show that this map restricts to a bijection between the tropical nef cone of $Y$ and the set of toric b-divisors that are b-Cartier and tropically nef. This provides a higher-dimensional generalization of Baker's specialization for curves and clarifies the birational nature of tropical line bundles. We also discuss the kernel of the map from line bundles to numerical tropical line bundles, which encodes the continuous moduli lost in tropicalization.