A journey to the Devil's Domain

 Tuesday, 18-April-2023 

Abstract

Max Noether said that algebraic curves were created by God and algebraic surfaces by the Devil. In this talk I will show some modern evidence supporting this statement. A moduli space is a parameter space for isomorphism classes of certain types of objects. Moduli spaces of curves have received a lot of attention for more than a century now. Discoveries about their properties have led to many applications. Interestingly, several of their properties were established before their existence was known. Mumford's Geometric Invariant Theory (GIT) brought an extremely powerful approach to moduli theory. In particular, using GIT, Mumford was able to prove the existence of those moduli spaces. In fact, together with Deligne, he also proved that there exist natural compactifications of these moduli spaces and that these compact moduli spaces are actually projective algebraic varieties. All of this was done in the 1960s and the expectation at the time was that this powerful theory would work for constructing and compactifying moduli spaces of higher dimensional varieties. It turns out that there are new difficulties in higher dimensions that GIT is not equipped to work with and eventually it took more than 5 decades of additional work to achieve the above goal.

In this talk I will explain one of the simplest issues we face in higher dimensions which forces us to reconsider how we approach the problem of moduli. I will also mention recent results that have finally completed the quest for the existence and projective compactification of moduli spaces of higher dimensional varieties.