Cone Theorem via Deligne-Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne-Mumford stacks. We also obtain a sharper bound on $-(K_X+D)$-degree of $(K_X+D)$-negative extremal rays for projective $\text{QQ}$-factorial log terminal threefold pair $(X,D)$.