Andres Fernandez Herrero (UPenn)
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PDL C-38
Title: Projectivity of the moduli of branchvarieties
Abstract: 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 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. This is used to obtain a moduli of "Noether normalized" varieties via classical GIT, which is birational to a moduli of K-semistable polarized varieties (indeed, the leading order term of the line bundle is the so-called CM line bundle).