Second Flip in the Hassett–Keel Program: Projectivity

For the moduli stacks of α-stable curves introduced in [AFSv15], we prove nefness of natural log canonical divisors generalizing a well-known result of Cornalba and Harris for the moduli stack of Deligne-Mumford stable curves. We then deduce the projectivity of the good moduli spaces of α-stable curves and identify these moduli spaces with the log canonical models of Mg,n appearing in the Hassett-Keel program.