Lifting tropical intersections

Understand

We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension.

• We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1.
• The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over non-noetherian valuation rings of rank 1.
• In particular, we prove subadditivity of codimension and a principle of continuity for intersections in smooth schemes over such rings, generalizing well-known theorems over regular local rings.