First Steps in Tropical Intersection Theory

TL;DR

This paper develops foundational aspects of tropical intersection theory, including defining cycles, divisors, and intersection products, with initial steps towards applications in enumerative geometry.

Contribution

It introduces the first parts of tropical intersection theory, defining cycles, Cartier divisors, and intersection products without rational equivalence, for fans and abstract cycles.

Findings

Established definitions of cycles and divisors in tropical geometry

Defined intersection products and push-forward/pull-back operations

Explored rational equivalence and cycle classes in R^n

Abstract

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.