| The subject of the talk will be a collection of projective varieties that contain the moduli space M_g of smooth curves of genus g as a dense open subset. The main focus will be on explaining how this set of models arises in three apparently rather different contexts: as modular compactifications (generalizing the Deligne-Mumford compactification by stable curves), as GIT quotients of pluricanonnical Hilbert schemes, and via the log minimal model program in birational geometry. I will provide a bit of basic background on each of these threads and then review the parallel progress in our understanding of them. |