Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8966119 | Journal of Pure and Applied Algebra | 2019 | 24 Pages |
Abstract
Let X and Xâ² be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product Xâ
Xâ² and the stable intersection trop(X)â
trop(Xâ²), when restricted to (the inverse image under the tropicalization map of) a connected component C of trop(X)â©trop(Xâ²). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between Xâ¾â
Xâ¾â²|Câ¾ and trop(X)â
trop(Xâ²)|Câ¾ via the compactified stable intersection, where the closures are taken inside the compactifications of T and Rn. In particular, when X and Xâ² have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that Xâ©Xâ² is allowed to be of positive dimension. Moreover, if Xâ¾â©Xâ¾â² has finitely many points which tropicalize to Câ¾, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Xiang He,