Lifting tropical intersections

TL;DR

This paper establishes conditions under which tropical intersection points lift to algebraic intersection points with expected multiplicities, extending classical intersection theory to tropical and non-noetherian valuation ring contexts.

Contribution

It introduces new lifting theorems for tropical intersections, utilizing advanced tropical geometry and finite type scheme analysis over non-noetherian valuation rings.

Findings

Points in tropical intersections lift to algebraic points with correct multiplicities.

Proves subadditivity of codimension in non-noetherian valuation ring schemes.

Establishes a continuity principle for intersections in smooth schemes over such rings.

Abstract

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.