Projectivity of the moduli of branchvarieties

It is a fact of life that the moduli of (reduced) equidimensional subvarieties of projective space is often not complete. In 2010, Alexeev and Knutson introduced a compactification called the moduli of branchvarieties. This is a proper Deligne-Mumford stack parameterizing equidimensional varieties equipped with a finite morphism to projective space. In principle, this is a great parameter space to carry out GIT constructions of moduli of varieties. However, there is one main caveat to this: Alexeev and Knutson left as an open problem whether their proper DM stack is projective. In this talk, I will explain a proof of projectivity obtained in joint work with Dan Halpern-Leistner, Trevor Jones and Ritvik Ramkumar.