Lifting non-proper tropical intersections

Details Lifting non-proper tropical intersections Lifting non-proper tropical intersections

2011-09-26

arxiv.org/abs/1109.5733v1

Mathematics

Algebraic Geometry

23 pages, 1 figure

Scientific paper

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.

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