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Lifting nonproper tropical intersections. (English) Zbl 1320.14078
Amini, Omid (ed.) et al., Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (ISBN 978-1-4704-1021-6/pbk). Contemporary Mathematics 605. Centre de Recherches Mathématiques Proceedings, 15-44 (2013).
Summary: 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 \(\mathrm{Trop}(X) \cap \mathrm{Trop}(X')\) 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.
For the entire collection see [Zbl 1281.14002].

MSC:
14T05 Tropical geometry (MSC2010)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14G20 Local ground fields in algebraic geometry
14G22 Rigid analytic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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