Cohomology bases of toric surfaces

Given a compact toric surface, the multiplication of its rational cohomology can be described in terms of the intersection products of Weil divisors, or in terms of the cup products of cohomology classes representing specific cells. In this paper, we aim to compare these two descriptions. More precisely, we define two different cohomology bases, the \emph{Poincar\'{e} dual basis} and the \emph{cellular basis}, which give rise to matrices representing the intersection product and the cup product. We prove that these representing matrices are inverse of each other.