A generalization of lifting non-proper tropical intersections

Let X and X′ be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product X⋅X′ and the stable intersection trop(X)⋅trop(X′), when restricted to (the inverse image under the tropicalization map of) a connected component C of trop(X)∩trop(X′). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between X‾⋅X‾′|C‾ and trop(X)⋅trop(X′)|C‾ via the compactified stable intersection, where the closures are taken inside the compactifications of T and Rn. In particular, when X and X′ have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that X∩X′ is allowed to be of positive dimension. Moreover, if X‾∩X‾′ has finitely many points which tropicalize to C‾, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).