Lifting nonproper tropical intersections

Abstract

ABSTRACT. 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 Trop(X)∩ Trop(X ′ ) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a suitable toric variety X(∆) and its associated extended tropicalization N R (∆); the algebraic intersection points lifting the stable tropical intersection will have tropicalization somewhere in the closure of C in N R (∆). The proof involves a result on continuity of intersection numbers in the context of non-Archimedean analytic spaces.

Non-Archimedean Coamoebae

“…This, as well as Propositions 1.2, 1.5, and 1.7 are the fundamental theorems of tropical geometry, and are due to many authors. For a discussion with references, see Section 2.2 of [15]. Another complete source presented in refreshing generality is [7].…”

Constructing polynomial systems with many positive solutions using tropical geometry

“…Then, to each such ξ , we associate a certain polynomial sub-system of (1.4), called reduced real systems with respect to ξ (see [8,Chapter 2.2.6]). All together, those reduced systems approximate all the non-degenerate parametrized solutions to (1.4) (see [11,17,18] and [5]). We adapt this approach to our setting by considering a particular type of parametrized non-degenerate solutions (α 1 (t), α 2 (t)), which we also call positive (i.e.…”

“…second) equation of (4.4). E. Brugallé and L. López De Medrano showed in [5, Proposition 3.11] (see also [11,17,18] for more details for higher dimension and more exposition relating toric varieties and tropical intersection theory) that the number of solutions of (4.2) with valuation at v 0 is equal to the mixed volume MV( ξ 1 , ξ 3 ) of ξ 1 and ξ 3 (recall that v 0 = ξ 1 + ξ 3 ). Since we assumed that (4.2) has only non-degenerate solutions in (K * ) 2 , we get MV( ξ 1 , ξ 3 ) distinct solutions of the system (4.2) in (K * ) 2 ( ξ 1 , ξ 3 ).…”

Intersection theory on tropicalizations of toroidal embeddings

“…It has its origin in tropical implicitization and states that push‐forward commutes with tropicalization. A version for the embedded case over fields with non‐trivial valuation has been proven in [, , ]. Theorem Let be a subtoroidal morphism of complete toroidal embeddings, and let .…”

“…This relation between intersection theory on toric varieties and tropical geometry has been extended by the development of tropical intersection theory on that incorporates the intersection rings of all normal toric compactifications of . Being able to intersect tropically as well as algebraically, it has been studied in how far tropicalization respects intersections or other intersection theoretic constructions, as, for example, push‐forwards ( in the case of non‐trivial valuations).…”