Tropical intersection products on smooth varieties

TL;DR

This paper develops a well-defined intersection product for tropical cycles on smooth tropical varieties, enabling the pull-back operation and extending classical intersection theory into tropical geometry.

Contribution

It introduces a new intersection product for tropical cycles on smooth varieties, ensuring well-defined cycles and facilitating pull-back operations.

Findings

Defines intersection product on tropical linear spaces

Extends intersection product to all smooth tropical varieties

Allows pull-back of tropical cycles along morphisms

Abstract

In analogy to chapter 9 of arXiv:0709.3705 we define an intersection product of tropical cycles on tropical linear spaces L^n_k, i.e. on tropical fans of the type max{0,x_1,...,x_n}^(n-k)*R^n. Afterwards we use this result to obtain an intersection product of cycles on every smooth tropical variety, i.e. on every tropical variety that arises from gluing such tropical linear spaces. In contrast to classical algebraic geometry these products always yield well-defined cycles, not cycle classes only. Using these intersection products we are able to define the pull-back of a tropical cycle along a morphism between smooth tropical varieties. In the present article we stick to the definitions, notions and concepts introduced in arXiv:0709.3705.