On birational geometry of the space of parametrized rational curves in Grassmannians

In this paper, we study the birational geometry of the Quot schemes of trivial bundles on ${\mathbb{P}}^1$ by constructing small $\mathbb{Q}$-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models which appear in the minimal model program on the Quot schemes. As a corollary, we show that the Quot schemes are Mori dream spaces and log Fano.