Secondary fans and Tropical Severi varieties

TL;DR

This paper investigates the complex relationship between tropical Severi varieties and secondary fans, providing conditions for when one is a subfan of the other and deriving a formula for intersection multiplicities.

Contribution

It introduces new criteria for when tropical Severi varieties are subfans of secondary fans and offers a partial converse, advancing understanding of their geometric relationship.

Findings

Identifies conditions preventing tropical Severi varieties from being subfans.

Provides a partial converse with conditions for containment of secondary fan cones.

Derives a combinatorial formula for tropical intersection multiplicities.

Abstract

This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi 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, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans.