Algebraic points, non-anticanonical heights and the Severi problem on toric varieties

In this article, we apply counting formulas for the number of morphisms from a curve to a toric variety to three different though related contexts (the first two are to be understood over global function fields): Manin's problem for rational points of bounded non-anticanonical height, asymptotics for algebraic points of bounded height and irreducibility of certain moduli spaces of curves, with application to the Severi problem for toric surfaces.