Moduli space of complete stable pairs

We define complete stable pairs on a smooth projective variety, and construct their moduli space. These moduli spaces have natural morphisms to the moduli of stable pairs and Quot-schemes. As an example, we show that the moduli of complete stable pairs on ${\mathbb{P}}^1$ is an iterated blowing-up of the moduli of stable pairs, similar to the construction of the space of complete collineations.