Lifting non-proper tropical intersections

TL;DR

This paper proves that stable tropical intersections of certain algebraic varieties over non-Archimedean fields can be lifted to actual algebraic intersection points, even when the tropical intersection component is positive-dimensional or unbounded.

Contribution

It establishes a lifting theorem for stable tropical intersections of subschemes in a torus, extending previous results to more general tropical components and using non-Archimedean analytic methods.

Findings

Stable tropical intersections lift to algebraic points with multiplicities.

Lifting may require passing to a suitable toric variety and extended tropicalization.

Continuity of intersection numbers is key to the proof.

Abstract

We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap Trop(X') lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(\Delta) and its associated extended tropicalization N_R(\Delta); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in N_R(\Delta). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.