These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel${}^{\text{'}}$fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.

We define an intersection product of tropical cycles on tropical linear spaces $L_k^n$, 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...