Lifting Tropical Intersections
By Brian Osserman, Sam Payne. 2010
98 citations recorded by the source corpus.
We show that points in the intersection of the tropicaliza- tions 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 intersec- tions inside an ambient subvariety of the torus, when the tropical- izations 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/3c5b2a7e4d22343d0c0f7bbd87b9f1eb6969bab1
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In Algebra and Number Theory
In Computational Theory and Mathematics
- Walter Gubler
- Erwan Brugallé
- Luc'ia L'opez de Medrano
- Joseph Rabinoff
- Shou-wu Zhang
- Matthew Baker
- Sam Payne
- Martin Ulirsch
In Geometry and Topology
- Walter Gubler
- Erwan Brugallé
- Joseph Rabinoff
- Shou-wu Zhang
- Matthew Baker
- Xinyi Yuan
- G. Mikhalkin
- Sam Payne
Citations in the corpus
- Inflection Points of Real and Tropical Plane Curves (2011, background): “d field K, and generalize to intersections of tropical varieties of higher dimensions. The existence of Γ can also be established using the fact that tropical intersections are tropical varieties (see [OP]). Using a more involved tropical intersection theory (see for example [Mik06], [S], or [A]), th”
- A guide to tropicalizations (2011, methodology): “ation. This means that we consider cycles up to the obvious linear extension of the above equivalence relation from prime components to all cycles. Note that the following result is a special case of [OP], Theorem 4.4.5. Here, we give a dierent proof using intersection theory with Cartier divisors. ”