Lifting nonproper tropical intersections
Brian Osserman, Joseph Rabinoff · Contemporary mathematics - American Mathematical Society · 2013
We prove that if X , X ′ X,X’ are closed subschemes of a torus T \mathbb {T} over a non-Archimedean field K K , of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C C of T r o p ( X ) ∩ \mathrm {Trop}(X)\cap T r o p ( X ′ ) \mathrm {Trop}(X’) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X ( Δ ) X(\Delta ) and its associated extended tropicalization N R ( Δ ) N_{\mathbb {R}}(\Delta ) ; the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C C in N R ( Δ ) N_{\mathbb {R}}(\Delta ) . The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.