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This article studies the relationship between tropical Severi varieties andsecondary fans. In the case when tropical Severi varieties are hypersurfacesthis relationship is very well known; specifically, in this case, a tropicalSeveri variety of codimension 1 is a subfan of the corresponding secondary fan.It was expected for some time that this continues to hold more generally, butKatz found a counterexample in codimension 2, showing that this relationship ismore subtle. The two main results in this paper are as follows. The firsttheorem finds a simple condition under which a tropical Severi variety cannotbe a subfan of the corresponding secondary fan. The second theorem provides apartial converse, namely, we find conditions under which a cone of thesecondary fan is fully contained in the tropical Severi variety. As a firstapplication of these results, we also find a combinatorial formula for thetropical intersection multiplicities for secondary fans.

Secondary fans and Tropical Severi varieties