The classic application is Deligne & Mumford's paper proving the irreducibility of the coarse moduli scheme $\overline{M}_g$ of stable genus g curves over any algebraically closed field. They proved irreducibility first for the moduli stack $\overline{\mathcal{M}}_g$ of curves, and then inferred from this result the irreducibility of the scheme.
user1504
- 6k
- 2
- 37
- 54