Lifting Non-proper Tropical Intersections

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) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(∆) and its associated Kajiwara-Payne extended tropicalization NR(∆); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in NR(∆). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.