Mathematics – Algebraic Geometry
Scientific paper
2010-07-08
Mathematics
Algebraic Geometry
42 pages, 6 figures; v2: minor changes including expository improvements, the addition of Remarks 4.2.6 and 4.3.2, and updated
Scientific paper
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. An appendix on the topology of finite type morphisms may also be of independent interest.
Osserman Brian
Payne Sam
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