Tropical lifting problem for the intersection of plane curves (Q6600952)

A common theme in tropical geometry is a super abundance of purely tropically defined objects. For example, one would expect two tropical plane curves \(\Gamma_1\) and \(\Gamma_2\) of degrees \(d_1\) and \(d_2\) to intersect in \(d_1d_2\) many points (counted with multiplicity). This is not true in a naive sense as tropical plane curves are 1 dimensional polyhedral complexes which may overlap in one-dimensional segments/ rays and hence have infinitely many points of intersection. In such a situation we pose the realizability question: \N\NFor which multiset \(D\) of \(d_1d_2\) many points in \(\Gamma_1 \cap \Gamma_2\) do there exist algebraic lifts \(C_i\) of the \(\Gamma_i\) such that \(D = \operatorname{trop}(C_1 \cap C_2)\)?\N\NA necessary condition for this was described in [\textit{R. Morrison}, Collect. Math. 66, No. 2, 273--283 (2015; Zbl 1331.14059)]: \(D\) must be linearly equivalent to the stable intersection divisor of \(\Gamma_1\) and \(\Gamma_2\). Using this, the present paper establishes a sufficient condition for realizability. There is no claim on this being necessary.\N\NThe main idea of the main theorem is as follows. Consider a bounded line segment \(L\) in \(\Gamma_1 \cap \Gamma_2\) such that the intersection multiplicity on \(L\) is 2. Then the stable intersection divisor is given by the two endpoints of \(L\) and anything linearly equivalent to that has to be supported on two points of some distance \(l\) from each of these endpoints. The sufficient condition says that \(l\) should be sufficiently large.