Second Flip in the Hassett-Keel program: Projectivity

Abstract

For the moduli stacks of α-stable curves introduced in [4], we prove nefness of natural log canonical divisors, thus 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.