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Tropical images of intersection points. (English) Zbl 1331.14059
Let \(K\) be an algebraically closed non-Archimedean field with a nontrivial valuation \(K^*\to\mathbb R\). The main example is the field of Puiseux series with complex coefficients. Consider two algebraic curves \(X\) and \(Y\) in the algebraic torus \((K^*)^2\). Assume that \(X\) and \(Y\) intersect in only finitely many points. The coordinatewise valuation map applied to the intersection points (with multiplicity) yields a divisor \(D\) inside the intersection \(\mathrm{Trop}(X)\cap \mathrm{Trop}(Y)\) of the tropicalizations of the two curves.
The main result of the paper under review is that \(D\) is linearly equivalent on \(\mathrm{Trop}(X)\) to the stable intersection divisor \(E\) of \(\mathrm{Trop}(X)\) and \(\mathrm{Trop}(Y)\). I.e., \(D-E\) is the divisor associated to a suitable tropical rational function on \(\mathrm{Trop}(X)\). Two proofs are given, one based on tropical modifications and one on Berkovich analytic spaces, although the latter proof requires an additional assumption on \(X\).
The author conjectures that his result is the only restriction on \(D\) (in a suitable sense). Detailed examples are given as evidence supporting this conjecture.

MSC:
14T05 Tropical geometry (MSC2010)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H50 Plane and space curves
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